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1 Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the dot product of v and w is given by v w v w + v w + v 3 w 3 For example, if v (4, 7, 6 and w (, 3, 9, then v w Notice that the answer is always a number Row Operations There are three types of row operations: ( Adding a multiple of one row to another ( Switching two rows (3 Multiplying a row by a constant There s a modification of ( that you can use to avoid fractions Namely, if you were going to do the operation row (3/7row, you could instead use 7row 3row This is just the first operation multiplied by the denominator 7 Any of these operations can be used to solve Ax b, and in finding inverses, but to find the LU factorization only use operations of type (and not the modification Echelon and Reduced Row Echelon (rref forms A pivot is the first nonzero entry in a row which has no other pivots directly above it You can identify the echelon form of a matrix by the following properties: ( Below each pivot are zeros ( Each pivot is to right of the one above it (3 Rows of all zeros (if any at all must come at the end Echelon form is like a lower triangular form for matrices which aren t necessarily square The reduced row echelon form of a matrix is an echelon form, but now each pivot must be a and there must be zeroes above the pivots as well as below them Echelon Forms RREF of above x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Elimination Matrices For an operation of the form rowi m rowj you get the elimination matrix E ij from the identity matrix by replacing entry (i, j (that s row i, column j with m If you do row operations in the order we ve done in class (use the pivot in the first row to get zeroes below it, then use the pivot in the second row to get zeroes below it, etc, then E ij will have its m in the same location as the entry that you were trying to zero out Note that if you did the operation rowi + m rowj, E ij would have a +m instead of m The matrix E ij has the effect that E ij times any matrix A subtracts m times row j from row i of A

2 LU Factorization If you do a sequence of row operations to reduce a matrix A into lower triangular form U, it can be written as something like E 3 E 3 E A U Solving this for A gives A (E E 3 E 3 U That big inverse is what we call L, and we find it similarly to finding the elimination matrices To find L, start with the identity matrix For each operation of the form rowi m rowj replace the (i, j entry with +m (note signs are the opposite of what they are for elimination matrices since L is an inverse of elimination matrices Again, if you do the operations in the order we ve done in class, then whatever position you re trying to get a zero in, the corresponding entry in L gets replaced by m For example let A Reduce A to lower triangular form, indicating the elimination matrices and find the LU Factorization of A E row row E row3 3row E L row3+4row U 3 4 Using LU Factorization To Solve Ax b We can write Ax b as LUx b Letting Ux c we see that we can solve Ax b in two steps: ( Solve Lc b ( Solve Ux c Each step is easy, only requiring back substitution and no row operations For example, use the LU Factorization in the above example to solve Ax b with b (, 5, 6 ( First solve Lc b c c 3 4 c 3 Write this as three equations c c + c 5 3c 4c + c 3 6 Solve to get c, c, c 3 4 ( Then solve Ux c 3 x x 4 x 3 Write this as three equations x + x + x 3 x + 3x 3 4x Solve to get x, x, x 3

3 PALU Factorization If you try to to the LU factorization and find that you can t do it without a row exchange, then use the P A LU factorization You start with row operations just like in the LU factorization, but as soon as you see you have to flip two rows, flip them instead in the original matrix and start over again with row operations on the this new A to get L and U Put your row flips into the matrix P You get P by starting with the identity matrix and flipping the same rows of it that you flipped during your row operations (ie, if you flipped rows and 3 during row operations, then P is the identity matrix with rows and 3 flipped For example let A 4 Find the PALU Factorization of A 4 row row At this point, stop and notice that the (, entry is This should be a pivot We need this to be nonzero in order to make the (3, entry zero, so we have to do a row exchange Exchange rows and 3 of A and start over 4 row row 4 row3 row The last matrix is in lower triangular form It is our U We have L U P Transposes and Symmetric Matrices To get A T, row i of A becomes column i of A T Remember the rules for transposes: ( (A T T A ( (A ± B T A T ± B T (3 (AB T B T A T (4 (A T (A T A symmetric matrix is a matrix whose entries are mirror images of each other on either side of the diagonal Mathematically, they are defined as matrices with A A T a u v w u b x y v x c z w y z d Example: If A and B are symmetric, show ABA is symmetric Answer: (ABA T A T B T A T ABA (using property (3 and the fact that A A T, B B T We have shown (ABA T ABA, so it is symmetric Example: If A and B are symmetric is AB(A + B symmetric? Answer: [AB(A + B T (A + B T B T A T (A T + B T B T A T (A + BBA Matrix multiplication is not commutative so this is not necessarily the same as AB(A + B So it is not necessarily symmetric For example, check that AB(A + B and (A + BBA are not the same with A ( B ( 3 3

4 Vector Spaces and Subspaces A vector space, roughly speaking, is a set where you can define addition and multiplication by a number in such a way that basic algebraic rules hold (like c(v + w cv + cw, and a few others Examples of vector spaces are all vectors with components, or all vectors with 3 components, etc Another example is the set of all x matrices (or 3x3 matrices, etc A subspace of a vector space is a set of some of the objects from the vector space which have to satisfy the following two properties: ( if v and w are in the subspace, then v + w has to be in it, too ( if v is in the subspace and c is a number, then cv must be in the subspace Usually the vectors in the set to be checked all have something in common or look a certain way To show it really is a subspace you have to check that both properties hold, in other words, check that v + w and cv have that same thing in common To show something is a subspace you have to use generic vectors, specific examples are not enough To show something is not a subspace however, specific examples are enough, just find a specific example where one of the properties fails Example: Show that all vectors (b, b, b 3 with b + b + b3 is a subspace of R 3 Answer: Examples of vectors in the subspace are (,, or (3,, These are vectors whose entries add up to This is what all vectors in the subspace have in common First check property Let c (c, c, c 3 and d (d, d, d 3 be any two vectors where c + c + c 3 and d + d + d 3 Then c + d (c + d, c + d, c 3 + d 3 For this to be in the subspace the entries in c + d must add up to This is true since (c + d + (c + d + (c 3 + d 3 (c + c + c 3 + (d + d + d 3 + Net check property Let c (c, c, c 3 be a vector with c + c + c 3 and let n be any number Then nc (nc, nc, nc 3 is in the subspace since nc + nc + nc 3 n(c + c + c 3 n Since both properties hold, it is a subspace Example: Is the set of all vectors (b, b, b 3 with all of the entries whole numbers a subspace of R 3? Answer: No You can check that the first property works since adding whole numbers gives whole numbers, but property doesn t work For example, pick the vector b (,, and let n 5 Then nb (5, 5, 5 is not in the subspace since its entries are not whole numbers 4

5 Nullspace The nullspace of a matrix A, denoted N(A consists of all vectors which are solutions to the equation Ax Example: Find the nullspace of the following matrix: A Answer: First reduce A to rref to get 3 A Keep in mind that finding the nullspace is the same as solving Ax, so rewrite the above matrix in equation form x x x 4 3x 5 x 3 + x 4 x 5 Solve these for the pivot variables x and x 3 x x + x 4 + 3x 5 x 3 x 4 + x 5 Now write x x x x 3 x 4 x 5 x + x 4 + 3x 5 x x 4 + x 5 x 4 x 5 x + x 4 + x 5 3 The vectors at the last step are what the book calls special solutions There is one for each free variable The null space consists of all linear combinations of these vectors Solving Ax b Solving Ax b consists solving Ax (ie finding the nullspace plus one more step First start by row reducing the augmented matrix For example let b ( 5, 5, and let A be the matrix in the nullspace example above Row reducing the augmented matrix to rref, we get: 3 Notice that if the last entry in the last row were not, then there would be no solution, as writing this in equation form, the last equation would be some nonzero number, which is nonsense Now to find the solution: First find the nullspace (which we did above and then find a particular solution A particular solution is the solution you get by setting the free variables (in this case x, x 4, x 5 all equal to What it works out to is just set the pivot variables equal to the entries in the last column in the augmented matrix, and set the free variables equal to zero Here we get x, x 3, x x 4 x 5 This give the particular solution (,,,, Adding this to the solution to Ax (the nullspace translates it into a solution to Ax b Thus the complete solution is: x + x + x 4 + x 5 3 5

6 Rank, etc A few definitions: ( The rank of a matrix is the number of pivots ( A matrix has full column rank if every column has a pivot (3 A matrix has full row rank if every row has a pivot (4 A pivot column is a column which contains a pivot To determine each of the above quantities, reduce the matrix to echelon form, from there it is easy Column Space The column space of a matrix A, denoted C(A, consists of all vectors b which are solutions of Ax b Another way to think of it is as all vectors which are linear combinations of the columns of A If A is simple enough you can determine C(A just by looking at it, but otherwise reduce A into echelon form to determine the pivot columns Then C(A is the set of linear combinations of the pivot columns of A (not the echelon form, but A itself Another way to determine C(A is to reduce the augmented matrix with the last column containing entries b, b, to an echelon form Here is an example: A An echelon form of A found by row reduction is From here we see that columns and 3 of A are pivot columns Thus C(A is all linear combinations of these columns In other words C(A consists of all vectors which can be written in the following form: C(A a + b where a and b are any numbers The other way we can find C(A is as follows: First, row reduce the following augmented matrix A row3 3row b b b row row b b b b 3 b 3(b b b b b b 3 row3 row b b b b 3 b For this to have a solution, the bottom right entry in the augmented matrix must be This means that b 3 3b b (setting the entry equal to and solving for b 3 Thus any vector in the column space is of the form b b 3b b Notice that any vector from the first way of finding C(A can be written in this form Full Row Rank and Full Column Rank If a matrix has full column rank (ie a pivot in every column, then the following are true: ( There are no free variables ( The null space of the matrix consists only of the vector of all zeros (3 Ax b has either no solution or exactly one solution If a matrix has full row rank (ie a pivot in every row, then the following are true: ( The number of free variables is equal to the number of columns minus the number of rows ( Ax b always has at least one solution 6

7 More About Ax b ( If the rref of a matrix has zero rows, then it s possible that there is no solution If the rref has no zero rows, then there is certainly a solution If there are free variables, then either there s no solution or there are infinitely many solutions (and nothing in between If there are no free variables, then there can t be infinitely many solutions There s either no solution or one solution in this case ( If A has full row rank and full column rank, then A is a square matrix (same number of rows as columns and the rref of A is the identity matrix Thus Ax b has exactly one solution, namely A b, and you can find it by row reducing A and back substitution (note that there s no nullspace to worry about This is the most important case (3 If A has full row rank and more columns than rows, then there are no zero rows in the rref, hence there is at least one solution However, since there are more columns than rows, there are some columns without pivots, hence there are free variables Thus there are infinitely many solutions (4 If A has full column rank with more rows than columns, then there are zero rows in the rref, thus it is possible there is no solution Since every column has a pivot, there are no free variables, hence there can t be an infinite number of solutions Thus there is either no solution or one solution (5 If A has neither full row rank nor full column rank, then the rref has zero rows and there are free variables Thus there is either no solution or infinitely many solutions (6 Remember, there can never be exactly or exactly 3 solutions Zero, one, or infinitely many solutions are the only possibilities (It s a good exercise to try to show why exactly solutions is impossible Linear Independence A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others Otherwise we say the vectors are linearly dependent Mathematically this is expressed by saying the vectors v, v,, v n are linearly independent if c v + c v + + c n v n only when c c c n This is the same as saying the matrix with these vectors as its columns has full column rank (do you know why? If a vector in the set can be written as a linear combination of others, for many applications it is somehow redundant A linearly independent set has no such redundancies, which is good If any of the following happens, then the vectors are linearly dependent ( One vector is a multiple of another, or there is an obvious way to combine some of the columns to get another one of the columns ( One of the vectors is the zero vector (3 There are more vectors than entries in each vector If the vectors pass the above three conditions, then create a matrix with the vectors as its columns Row reduce the matrix to echelon form If every column has a pivot, then the vectors are linearly independent Otherwise they are linearly dependent Examples: Let v v 3 v v 4 ( {v, v 3 } are linearly dependent since v 3 is twice v ( {v, v 3, v 4 } are linearly dependent since v is the zero vector v v v 7 (3 {v, v 4 } are linearly independent To see this, make a matrix whose columns are v and v 4 and row reduce it to an echelon form The echelon form has pivots in every column 3 (4 {v, v 4, v 5, v 6 } are linearly dependent because there are 4 vectors, but only 3 entries in each (5 {v, v 4, v 6 } are linearly dependent To see this, make a matrix whose columns are v, v 4, and v 7 and row reduce it to an echelon form The echelon does not have pivots in every column

8 Math 43 Review Notes - Chapter 4 Orthogonal Subspaces Recall that two vectors v and w are orthogonal (ie perpendicular if their dot product v w is If we think of v and w as arrays with only one column, then we can also write the dot product of v and w as a multiplication of matrices, namely v T w You might see this from time to time in the book We say that two subspaces V and W of a vector space are orthogonal if every vector in V is orthogonal to every vector in W To determine if two subspaces are orthogonal, it is enough just to check that each vector in a basis for V is orthogonal to each vector in a basis for W Here is an important example of orthogonal subspaces: Example: The nullspace and row space of a matrix A are orthogonal subspaces Remember, vectors in the nullspace of a matrix A are vectors x for which Ax, and vectors in the row space are linear combinations of the rows of A A small example is enough to understand why the subspaces are orthogonal Let A ( Let x (x, x, x 3 be in the nullspace of A Then Ax Write this out in equation form: x + 3x + 4x 3 5x + 6x + 7x 3 Looking closely at these, we see the first equation is just the dot product of row and x, and the second is the dot product of row and x Since these are zero, the equations are saying that row and row are orthogonal to x So any combination of the rows is also orthogonal to x Therefore we have shown that the row space and nullspace are orthogonal Orthogonal Complement The orthogonal complement of a subspace V consists of all vectors which are orthogonal to each vector in V It is denoted by V Like before, to find V it is enough to find all vectors which are orthogonal each vector in a basis for V Example: Let V be the subspace which consists of all vectors x (x, x, x 3 which satisfy x + 3x + x 3 (a plane in R 3 Find V Solution: First find a basis for V Solve the equation defining V to get x 3x x 3 Then any x in V can be written as x x x 3x x 3 x x 3 + x 3 x 3 x 3 So ( 3,, and (,, are a basis for V (You can check for yourself that they are linearly indendent Any vector b (b, b, b 3 in V is orthogonal to both of these vectors In other words: (b, b, b 3 ( 3,, (b, b, b 3 (,, 3b + + b 3 b + b + b 3 3b b b Therefore, plugging these into b we get b b b b b b b 3 3b 3 So, V consists of all multiples of the vector (,, 3 (V is a line in R 3, perpendicular to the plane x + 3x + x 3 Example: Not only are the row space and nullspace orthogonal, but in fact they are orthogonal complements of each other Try to see why

9 Make sure you understand the difference between the definitions The first definition is about a relationship between two subspaces: V and W are orthogonal if every vector in V is orthogonal to every vector in W Notice also that the only vector that can be in both V and W is, since it is the only vector whose dot product with itself is (ie it is the only vector which is perpendicular to itself The second definition gives a new subspace V, the orthogonal complement, which contains every single vector which is orthogonal to all the vectors in V Projections First we will look at projecting a vector b onto another vector a One way to think about a projection is as the shadow that b casts on a Another way is to think of it as asking how much of b is in the direction of a The projection of b onto a is given by p a b a a a The fractional term gives the percentage or fraction of a that the shadow of b takes up; in the book it is called x Example: Project b (, 4, onto a (3,, 5 and c (3,, Solution: Projecting b onto a we get (, 4, (3,, 5 p (3,, 5 (3,, Projecting b onto c we get p (, 4, (3,, (3,, (3,, 3 4 3/ 5/ 3 The vectors b and c are perpendicular (since their dot product is, hence the projection of b onto c is just, as we would expect There s one other way to think about projections Instead of just projecting b onto a, think about it as is projecting b onto the line through a Then p is the closest vector to b which lies on the line The vector e in the picture is the error between b and p It is just b p Since a line is just a special kind of subspace, this thinking gives us a way to project a vector b onto any subspace V The projection will be the vector in V which is closest to b We will consider projecting onto the column space of a matrix A In other words, we are looking for the vector which is a linear combination of the columns of A which is closest to b To do this we use the following: P A(A T A A T p P b The matrix P is called the projection matrix It will project any vector onto the column space of A

10 Example: Let A Find the projection matrix P, and the projection of b (,, 3 onto the column space of A Solution: Compute: A T A ( ( (A T A ( 3 A(A T A ( 3 3 P A(A T A A T ( 3 p P b 3 3 Orthogonal Bases Recall that vectors are orthogonal if their dot product is If in addition to being orthogonal the vectors are all unit vectors (ie have length, then we say the vectors are orthonormal A convenient way to say this is that the vectors v, v,, v n are orthonormal if v i v j unless i j, in which case it equals Example: Let v 3, v 6, v It is easy to check that v v v v v 3 v 3, and v v v v 3 v v 3 So the vectors are orthonormal The Gram-Schmidt Process We say that a basis is an orthogonal basis if its vectors are orthogonal, and is an orthonormal basis if its vectors are orthonormal For many applications orthogonal and orthonormal bases are much easier to work with than other bases (Remember that a space has many different bases The Gram-Schmidt process gives a way of converting a basis into an orthogonal or orthonormal basis You start with n linearly independent vectors x, x,, x n and Gram-Schmidt will create orthogonal vectors v, v,, v n which span the same space as the x s In other words, Gram-Schmidt takes a basis for a subspace and turns it into an orthogonal basis for the same subspace The idea relies on projections Start by letting v x Then let p be the projection of x onto v Notice that x p is orthogonal to v So let this be v Notice that v and v span the same space as x and x Now consider the projection p of x 3 onto the subspace spanned by v and v Like before, x 3 p is orthogonal to both of v and v, and since v and v are orthogonal, it has a simple formula (no messy matrix calculations Continue projecting to get the rest of the v s 3

11 Here is the process: v x v x x v v v v v 3 x 3 x 3 v v v v x 3 v v v v v 4 x 4 x 4 v v v v x 4 v v v v x 4 v 3 v 3 v 3 v 3 and so on To make this set of vectors orthonormal, the vectors need to be unit vectors Remember how to do that, we divide each vector by its length, ie our orthonormal vectors are v v, v v, v 3 v 3, Example: Find a set of orthonormal vectors which span the same space as x (,,, x (,, 3, and x 3 (4, 3, 8 Solution: Check for yourself that the given vectors are linearly independent Now use Gram-Schmidt to find the corresponding orthogonal vectors v, v, and v 3 v v 3 v To make v,v, and v 3 orthonormal, divide each by its length to get 3 6,, Update: (/3/5 An easier way to find V This method uses the fact that the row space and nullspace of a matrix are orthogonal To find V, first find a basis for V Next create a matrix A from the basis vectors by letting them be the rows of A Then V is the nullspace of A In fact, this is really very similar to the method on the first page 4

12 Math 43 Review Notes - Chapters 3, 5, & 6 A basis for a vector space is a collection of vectors that satisfy: The vectors are linearly independent (No redundant vectors Every element in the vector space is a linear combination of the basis vectors These say that a basis is a set from which you can obtain every vector in the vector space, and that this set is as small as possible Many properties of a vector space can be proved just by proving them for the basis vectors A typical basis has only a few vectors, whereas the vector space probably has infinitely many, so this is a big help Example: The vectors (, and (, are a basis for R, the set of all vectors with two components It is easy to see that they are linearly independent, and any vector with two components can be written in terms of them For example, [ 7 5 [ 7 [ 5 and in general, [ a b [ a [ + b Similarly, (,,, (,,, and (,, are a basis for R 3 Can you find a similar basis for R n, the set of all vectors with n components? A space has many different bases For example, given a basis, just multiply each basis vector by a constant to get a new basis In the above example, for instance, (, and (, are also a basis for R Note however that all the bases for a space have to have the same number of vectors This number is called the dimension of the space Example: Find a basis for the subspace of R 3 consisting of all vectors whose components add up to Solution: Let b (b, b, b 3 be any vector in the subspace Then the sum of its components is, ie, b + b + b 3 Solve this for b to get b b b 3 Now write b b b b b 3 b b + b 3 b 3 b 3 We have just shown that any vector in the subspace can be written as a linear combination of the vectors (,, and (,,, and you can easily check that they are linearly independent, thus these two vectors are a basis for the subspace Since there are two vectors in the basis, the dimension is of the subspace is Often we will have a set of vectors that we know satisfies property ( of a basis, ie any element in the space is a combination of the vectors We call such a set a spanning set, or we say it spans the vector space Suppose we are given a set of vectors and we want to find a basis for the set they span They automatically satisfy property ( of a basis, but not necessarily property (, since there may be redundant vectors To get rid of the redundant vectors we can make a matrix with the vectors as its columns and row reduce to find the pivot columns The pivot columns correspond to the vectors we keep for our basis, while the other columns correspond to the redundant vectors which we will throw away Example: Find a basis for the set spanned by the vectors (,, 3, (, 4, 4, and (,, Solution: As we said above, property ( is automatically satisfied here, so we just have to throw out the redundant vectors Following the above discussion, The first two columns are the pivot columns, hence the corresponding vectors (,, 3 and (, 4, 4 are a basis Note that depending on which row operations you do, you might get a different answer But that s ok, as there are many possible bases for a space

13 Example: Do the vectors (,,, (,, 3, and (, 4, 9 form a basis for R 3? Solution: We have to verify properties ( and ( here We can do this in one big step Row reduce the matrix with the vectors as its columns If there is a pivot in every column, then the vectors are linearly independent If there is a pivot in every row, then the vectors span R 3 (This is true since having a pivot in every row means that Ax b has a solution for any b, which means that any vector b in R 3 can be written as a linear combination of the columns of A There is a pivot in every row and every column, so the vectors do form a basis Bases for Nullspace, Row Space and Column Space Let A be a matrix with echelon form U and rref R Nullspace The nullspace of A consists of all vectors x for which Ax To find it, reduce A to rref R A 3 6 R Write Rx in equation form and solve for the pivot variables x + x + 3x 4 x 3 + 4x 4 x 5 x x 3x 4 x 3 4x 4 x 5 Use this to find an expression for a typical vector in the nullspace x x 3x 4 3 x x x 3 x 4 x 4x 4 x 4 x + x 4 4 x 5 So any vector in the nullspace can be written as a combination of the vectors (,,,, and ( 3,, 4,, It turns out that the vectors one gets at this point are always linearly independent So these two vectors are a basis In general, then the vectors one gets at the last step are a basis Notice that there is one vector for each free variable, so the dimension of the nullspace is equal to the number of free variables (ie the number of non-pivot columns In this example the dimension is Column Space The column space of a matrix A consists of all vectors which are linear combinations of its columns Another way to think of it is as all vectors b for which Ax b has a solution The columns of A are a spanning set, so to get a basis we need to throw out any redundant vectors To do this we row reduce A to an echelon form to see which columns have pivots (any echelon form will do, rref is ok, but not necessary The pivot columns of A are a basis, and the dimension of the column space is the number of pivots For example, A U Columns,3, and 5 are the pivot columns, so they are a basis, ie (,,, ( 3,, 3, and (4, 6, are a basis Make sure to use the columns of A itself The dimension in this case is 3

14 Row Space The row space of a matrix A consists of all vectors which are linear combinations of its rows Another way to think of it is as the column space of A T The rows of A are a spanning set, so to get a basis we need to throw out any redundant vectors To do this we row reduce A to an echelon form to see which rows have pivots (as before, any echelon form will do The pivot rows of A (or U or R form a basis The dimension of the row space is the number of pivots For example, A U Rows,, and 3 are pivot rows, so they are a basis, ie, (,, 3, 9, 4, (,,, 3, 6, and (, 4, 3, 6, are a basis Notice here that you could instead use the pivot rows of U or R as a (nicer basis The dimension in this case is 3 Remarks Row operations don t change the nullspace or the row space; that s why we can use the pivot rows of A, U, or R as a basis for the row space However, row operations do change the column space, so it s important that we only use the pivot columns of A itself as a basis for the column space of A Notice also that the dimensions of the row space and the column space are always the same number r, and r plus the dimension of the nullspace equals the number of columns of A Determinants The determinant of a matrix is a single number that contains a lot of information about the matrix The determinant of A is denoted either by det(a or A and it only applies to square matrices (matrices with the same number of rows as columns We have three different ways of finding determinants The best way is often to use a combination of the three Method : Row reduction ( The determinant of a triangular matrix (all zeroes above or below the diagonal is the product of the entries on the diagonal ( Switching two rows switches the sign of the determinant (3 Multiplying a row by a number multiplies the determinant by that number, so you ll have to divide your answer by that number to cancel it out (4 Adding or subtracting a multiple of one row from another doesn t affect the determinant (5 Row operations where the row you replace is also multiplied by a number do change the determinant (say you replace row with 3 row row, for example If you replace row j with m row j - n row k, then this multiplies the determinant by m, so you have to divide your answer by m to cancel out its effect Combining these gives us a method for finding determinants Row reduce the matrix to a triangular form, and then multiply the entries on the diagonal Make sure to take into account signs from row switches, and be sure to divide by any multiples you introduced if you do the row operations in (3 or (5 Example: Find the determinant of Solution: Switch rows and 3 The resulting matrix is triangular The minus sign comes from the row switch ( 3

15 Example: Find the determinant of Solution: Row reduce to a triangular matrix r r 3 r3 r r3 r 3 5 The row operations are of the type in (4 that doesn t affect the determinant Example: Find the determinant of Solution: Row reduce to a triangular matrix r r Another way to do this using the row operation in (5 would be r 7r r r r If you don t mind the fractions, then you could also do this problem just by subtracting (7/3row from row to reduce to triangular form This has the benefit of not having any multiples to remember to cancel out Method : Big Formula The determinant can be given by a formula just in terms of the entries of the matrix It is most useful when the matrix is 3 3 or smaller, since for an n n matrix the formula has n! terms (For instance, when n 6, that s 7 terms ( A matrix only has one entry The determinant is equal to that entry ( For a matrix, the formula is a b c d ad bc (3 For a 3 3 matrix the formula is a a a 3 a a a 3 a 3 a 3 a 33 a a a 33 + a a 3 a 3 + a 3 a a 3 a a 3 a 3 a a a 33 a 3 a a 3 This is already a lot to remember, but there is a shortcut which only works in the 3 3 case 4 Example: (Shortcut for 3 3 Compute First copy the indicated entries onto the right side of the matrix Multiply the entries along each indicated diagonal, and add up the three values you get This gives the first three terms in the big formula (((6 + (4(7( + ((3(5 7 Next copy the other indicated entries to the left side of the matrix Multiply the entries along each indicated diagonal, take the negative of each product, and add up the three values you get This gives the last three terms in the big formula The answer is then ((( (4(3(6 ((7(5 4 4

16 (4 There is a formula which works for bigger matrices If you look at the 3 3 formula, you ll see that for every possible permutation (x, y, z of (,, 3, there s a term a x a y a 3z (For example, (3,, and (,, 3 are possible permutations; there s 6 total Whether we add or subtract a given term is determined by how many steps it takes us to get from (,, 3 to (x, y, z An even number of steps corresponds to adding, and an odd number corresponds to subtracting (For example, it takes 3 steps to get from (,, 3 to (3,, (,, 3 (3,, (3,, To get the 4 4 formula, use permutations of (,, 3, 4 instead of (,, 3 There should be 4 terms, one for each possible permutation (For instance, a 3 a 4 a 3 a 4 is the term corresponding to the permutation (3, 4,, Bigger matrices work similarly However, with so many terms, the big formula is usually not practical for matrices larger than 3 3 Method 3: Cofactors Let M ij be the matrix left over after crossing out row i and column j The term C ij ( i+j det(m ij is called a cofactor We can use cofactors to compute the determinant The idea is to pick any row or any column of the matrix, and for each entry in that row or column, multiply the entry and its cofactor, then add up all of the products you computed In terms of formulas, det A a i C i + a i C i + + a in C in if you expand across row i, det A a j C j + a j C j + + a nj C nj if you expand across column j The ( i+j in C ij gives either a plus or minus sign One way to get the signs right is to remember the sign matrix ( , or for 4 4, For any size matrix, start with a + in the first entry, and the signs + + alternate from there The above formulas may not be very enlightening; the method is best demonstrated with some examples 3 4 Example: Compute the determinant of 5 three ways, 6 (a by expanding across row, (b by expanding down column, (c by expanding across row 3 Solution: (a The formula says det A a C + a C + a 3 C 3 We know a, a 3, a 3 4, so we just have to find the cofactors 3 4 ( A 5 Signs + M 5 M M det A ( 3(6 + 4( 6 Use the formula ( from method for evaluating each determinant Streamlining things a bit, we see each term consists of two parts, first an entry in row, added or subtracted according to the corresponding entry in the sign matrix (or by the formula ( i+j, and second, the determinant of the matrix we get by crossing out the row and column containing the entry 5

17 (b ( det A (6 + ( ( 4 6 (c ( det A (5 8 ( 4 + 6(4 3 6 We didn t have to write the last two terms in (c since they were multiplied by As you can see, the easiest of the three to compute is (c When evaluating a determinant by cofactors, you get to pick the row or column to expand across Usually it is easiest to expand across the one with the most zeros The Fastest Way: Probably the fastest way to compute determinants, especially 4 4 determinants, is to combine the methods Use row operations to create a row or a column with a lot of zeroes, and then use cofactor expansion down that row or column Example: Find the determinant of Solution: Subtract row from row and subtract row from row 3 These operations don t change the determinant Then expand down column We get r r r4 r Expand Now compute the 3 3 determinant using any method The answer is 4(7 68 Properties of Determinants ( det A A has no inverse In other words: if det A, then you know A has no inverse, and conversely, if you know A has no inverse, then it must be true that det A ( det AB (det A(det B (3 det A / det A Try to prove this in one line using property ( and the fact that AA I (4 det A T det A This rule implies that column operations affect the determinant in the same way that row operations do (Column operations are like row operations, except that you perform them on the columns instead of the rows 6

18 Cramer s Rule Cramer s Rule is a method for solving Ax b using determinants Since it uses determinants, it is really only practical for hand computations on small matrices, however it is useful for proving things, as well as for hand computations when row operations would involve a lot of fractions Method: ( Compute det A If you get, stop Cramer s rule won t work ( Let B be the matrix you get by replacing column of A with b Let B be the matrix you get by replacing column of A with b, etc (3 Then the solution is given by x det B det A, x det B det A, x 3 det B 3 det A, Example: Use Cramer s rule to solve Ax b where A Solution: Compute the following: and b 3 6 det A det B det B det B Therefore x 5, x 7, x 3 4 Inverses by Cofactors: We can regard finding A as solving the equation AA I for A We can convert this into a system of equations of the form Ax b which we can solve by Cramer s Rule In the end we get the following method for finding A Just as with Cramer s rule, it s usually not practical for hand computations on large matrices, but it is useful for theoretical calculations and hand computations on and 3 3 matrices, especially when row operations involve lots of fractions Method: ( Compute det A If you get, then stop as there is no inverse ( Compute all the cofactors and put them into a matrix C, so that C ij is in row i, column j (3 A is given by C T / det A One way to do step is to compute all the determinants you get by crossing out a row and a column of A, and put them into a matrix The determinant calculated by crossing out row i and column j goes into row i, column j of this new matrix Then get the signs right by using the sign matrix Don t touch entries which are in the same location as the + signs of the sign matrix However, do change the sign of entries which are in the same location as the signs of the sign matrix 7

19 Example: Find the inverse of Solution: ( Compute det A to get 5 3 ( Compute all the determinants you get by crossing out a row and a column of A, and put them into a matrix Now apply the sign matrix C (3 Finally, transpose C and divide by det A A Eigenvalues and Eigenvectors Consider multiplying a vector x by a matrix A The resulting vector Ax most likely is of a different length and points in a different direction than x Those vectors x for which Ax points in the same direction as x are called eigenvectors Essentially this means that multiplication by A on x acts just like multiplying x by a constant If we call the constant λ, then we get the equation Ax λx if x is an eigenvector The constant λ is called an eigenvalue How to Find Eigenvalues and Eigenvectors The equation Ax λx tells us how to find the eigenvalues and eigenvectors Subtract λx from both sides and factor it out to get (A λix The only way for there to be nonzero eigenvectors (ie interesting eigenvectors is if A λi is not invertible Recall that this is true when det(a λi Thus to find the eigenvalues we solve the polynomial equation given by det(a λi For each λ found above there are different eigenvectors The eigenvectors x satisfy (A λix, ie they are elements of the nullspace of A λi Note that A λi has the same entries as A except that the diagonal entries have λ subtracted from them For example: ( ( 4 λ 4 A A λi λ 8

20 In summary, To find the eigenvalues, compute det(a λi This is a polynomial equation Set it equal to and solve for λ To find the eigenvectors for an eigenvalue λ, find the nullspace of A λi The vectors in the nullspace are the eigenvectors corresponding to λ Do this for all the eigenvalues Example: Find the eigenvalues and eigenvectors of Solution: First find the eigenvalues: ( λ 3 λ (4 λ( λ 3 λ 6λ + 5 (λ (λ 5 So λ, 5 are the eigenvalues Eigenvectors for λ : Find the nullspace of A 3I ( ( ( 3 3 (/3r r r A 3I x [ x x [ x x x [ Therefore the eigenvectors for λ are all multiples of (, Eigenvectors for λ 5: Find the nullspace of A 5I ( ( ( 3 r3+r 3 r 3 A 5I 3 x [ x x [ 3x x x [ 3 Therefore the eigenvectors for λ 5 are all multiples of (3, Example: Find the eigenvalues and eigenvectors of Solution: x + x x x x 3x x 3x First find the eigenvalues: det(a λi might look imposing, but use row operations to simplify it, and then expand across row λ λ λ r r3 r r3 λ λ λ λ λ λ λ λ λ + λ λ λ( λ( λ λ + λ( λ 4λ λ 3 + λ λ (6 λ Thus the eigenvalues are λ, 6 Eigenvectors for λ : Find the nullspace of A I A I rref x + x + x 3 x x x 3 x x x x 3 x x 3 x x 3 x + x 3 Therefore the eigenvectors for λ are all linear combinations of (,, and (,, 9

21 Eigenvectors for λ 6: Find the nullspace of A 6I A 6I 4 4 rref x x 3 x 4 x 3 x x x x 3 x 3 x 3 x 3 x 3 Therefore the eigenvectors for λ 6 are all multiples of (,, x x 3 x x 3 Remarks about Eigenvalues and Eigenvectors: There are a few ways to check your work: The product of the eigenvalues equals det A The sum of the eigenvalues equals the sum of the diagonal entries of A 3 Eigenvalues are numbers which make A λi not invertible Therefore, if you find that the only eigenvector you get is (or equivalently, if you have a pivot in every column, then there is certainly a mistake somewhere The eigenvalues of a triangular matrix are the entries on the diagonal It s possible that the all the eigenvalues are imaginary numbers This means that the only (real eigenvector is Finding the eigenvalues of a n n involves solving a polynomial of degree n which is often difficult for n > Diagonalizing a Matrix A diagonal matrix is a matrix whose entries above and below the diagonal are all zero Diagonal matrices look like the identity matrix, except that the entries on the diagonal don t have to be all ones Diagonal matrices are nice to work with because they have so many zero entries Using eigenvalues and eigenvectors, we can rewrite a matrix in terms of a diagonal matrix To do this, ie to diagonalize an n n matrix, we need the matrix to have n linearly independent eigenvectors, otherwise we can t do it Example: ( If all the eigenvalues of a matrix are different, then there will be n linearly independent eigenvectors, so the matrix is diagonalizable ( Suppose a 3 3 matrix has eigenvalues 3 and 4, and in finding the eigenvectors you find for λ 3, the nullspace of A λi is given by x x (,, + x 3 (,, and for λ 4, the nullspace of A λi is given by x x 3 (3,, The vectors (,,, (,,, and ( 3,, are three linearly independent eigenvectors, so the matrix is diagonalizable (3 Suppose in the above example, the nullspace of A λi for λ 3 didn t have the second term Then we could only find two linearly independent eigenvectors, and so we couldn t diagonalize the matrix When you find the nullspace of A λi, the vectors at the last step are a basis for the nullspace (For example, in (, the vectors we re referring to are (,, and (,, It turns out that the set of all the basis vectors for all the eigenvalues is linearly independent So you just have to count up the total number of basis vectors you find, and if the total is n, then the matrix is diagonalizable, otherwise it is not

22 How to Diagonalize A Matrix Let Λ be the diagonal matrix whose entries along the diagonal are the eigenvalues of A Let S be the matrix whose columns are the eigenvectors of A It is important that the order of the eigenvectors in S corresponds to the order of the eigenvalues in Λ For example, if the eigenvalue λ 4 is in column 3 of Λ, then the eigenvector in column 3 of S must be an eigenvector you found using λ 4 We can diagonalize A by writing it as A SΛS Example: Diagonalize A ( 5 Solution: The matrix is upper triangular so the eigenvalues are λ, 5, the entries on the diagonal ( ( [ [ [ rref x x λ : A λi x 3 x x x ( 3 λ 5 : A λi ( ( Λ S 5 3 ( rref /3 S 3 x x /3 x ( 3 [ x x [ [ x /3 /3 x x Notice that for the second column instead of using (/3, we used (, 3 This is ok since the eigenvectors for λ 5 are all the multiples of (/3, (, 3 is such a multiple, and we chose it since it has no fractions This wasn t necessary, but it gives a nicer S So we diagonalize A as A SΛS 3 ( 3 ( 5 ( 3 Using Diagonalization to Find Powers of A Suppose A is diagonalizable, so that A SΛS Observe the following: A (SΛS (SΛS SΛ S A 3 A A (SΛ S (SΛS SΛ 3 S We used the fact that SS I to simplify both expressions In general we get A k SΛ k S This is useful because raising diagonal matrices to powers is particularly simple just raise each diagonal entry to the power This doesn t usually work with non-diagonal matrices Thus to compute A k we only need three multiplications instead of k + 3 multiplications Example: Use the diagonalization of the matrix in the example above to compute A k Solution: A k SΛ k S 3 ( 3 ( k 5 k ( 3 ( k 5 k k ( 3 ( k (5 k k /3 5 k We just multiplied all the matrices together Notice that we brought the /3 inside at the last step

23 Math 43 Review Notes - Chapters 6 & 83 Symmetric Matrices and Orthogonal Diagonalization A symmetric matrix has only real eigenvalues Its eigenvectors can always be chosen to be orthonormal A symmetric matrix can always be diagonalized, unlike other matrices When we diagonalize a symmetric matrix we get a special diagonalization called an orthogonal diagonalization We diagonalize A as A QΛQ T, where Λ is the diagonal matrix with the eigenvalues of A on the diagonal, and Q is the matrix whose columns are unit eigenvectors Make sure that the unit eigenvectors in Q line up with the corresponding eigenvalues in Λ This diagonalization is very similar to the usual SΛS diagonalization The difference here is that we use unit eigenvectors, and since the columns of Q are orthonormal, Q Q T, so we don t have to compute an inverse ( 8 6 Example: Orthogonally diagonalize 6 8 Solution: First find the eigenvalues 8 λ λ λ (λ (λ + Thus the eigenvalues are and - Now find the eigenvectors and unit eigenvectors ( ( [ [ 6 rref 3 3 3/ λ : is an eigenvector, and 6 8 / is the unit eigenvector λ : Finally, we write ( A QΛQ T rref ( /3 ( 3/ / / 3/ [ 3 ( is an eigenvector, and ( 3/ / / 3/ [ / 3/ is the unit eigenvector Markov Matrices A Markov matrix is a matrix whose entries satisfy: All the entries are The entries in each column add up to ( ( 5 /3 For example, and are Markov matrices 8 5 /3 A Markov matrix always has as an eigenvalue The eigenvectors corresponding to λ are called steady-state eigenvectors To see why they are called this, recall the equation Ax λx that defines eigenvalues and eigenvectors With λ this says Ax x; in other words multiplication by A doesn t change x Many real-life phenomena are modelled by the equation x n+ Ax n, with A a Markov matrix The long range behavior of such a system is determined by the steady state eigenvectors Example: Find the eigenvalues and a steady-state eigenvector for A ( Solution: We know right away that is an eigenvalue for A, since A is a Markov matrix To find the other, remember that the sum of the eigenvalues is equal to the sum of the diagonal entries of A So we solve + λ to get the other eigenvalue The steady-state eigenvectors are eigenvectors for λ : ( ( 8 5 rref 5/8 A λi 8 5 [ 5/8 Thus the steady state eigenvectors are multiples of [ 5 For instance, 8 or [ 5/3 8/3 are two examples If one eigenvalue gives more than one linearly independent eigenvector, then you will have to orthogonalize the vectors using Gram- Schmidt or something else We didn t consider anything like this in class, however, so don t worry about it

24 More about Steady-State Eigenvectors Markov matrices are useful for modelling many things Below is a simple example Example: In any year, 9% of deer in the forest remain there, while 8% find their way into the suburbs (and people s backyards, where they eat their shrubbery In addition, 88% of the deer in the suburbs remain there, while % are caught and returned into the forest (Note that this is a simplified example; it doesn t take into account a lot of factors Can you think of any assumptions in this model? Let f n and s n denote the number of deer in the forest and suburbs, respectively, in year n Then we can write the above paragraph mathematically as f n+ 9f n + s n s n+ 8f n + 88s n We can write this system of equations in matrix form as [ ( [ fn+ 9 fn s n s n Let s call the matrix A It is a Markov matrix, and thus it has λ as an eigenvalue The steady-state eigenvectors (eigenvectors for λ are found by computing the nullspace of A λi ( ( 8 rref 3/ A λi 8 Thus the steady state eigenvectors are multiples of (3/, One multiple without fractions is (3, To express this in terms of percents, divide each entry by the sum of the two entries to get (3/5, /5 (6, 4 Thus we expect that after many years (ie, when n is large 6% of the total deer population will be in the forest and 4% will be in the suburbs To see precisely why this is true, notice from the matrix equation above that [ [ [ [ ( [ [ f f f3 f f A and A A A A f s s s 3 s s s In general, we see that [ fn+ s n+ A n [ f s So the population is closely related to A n Recall that we can use diagonalization to find A n We diagonalize A as A SΛS, and from there we get A n SΛ n S To diagonalize A we need the other eigenvalue and its eigenvector As in the previous example, to find the other eigenvalue we solve λ to get λ 8 An eigenvector for λ 8 turns out to be (, Thus we can write ( ( ( A n SΛ n S 3 8 n ( 3 + (8 n 3 3(8 n (8 n + 3(8 n The last equality comes from multiplying the three matrices together Notice that for fairly large values of n, 8 n is very small For example, and So after a number of years we can ignore the 8 n term and say A ( 3/5 3/5 /5 /5 Notice that the columns are both the steady-state eigenvector of A that we found above Moreover, [ [ ( [ [ 3 fn+ A n f 3/5 3/5 f 5 f s [ 3/5 s n+ s /5 /5 s 5 f + 5 s (f + s /5 So we see that in the long run (in this case after roughly -5 years 3/5 (6% of the deer population will be in the forest and /5 (4% in the suburbs In addition, the equation above says that the percentages will remain the same for all later years This is why we call the vector (3/5, /5 a steady-state Finally, notice that the percentage of deer in the forest versus the suburbs during year had absolutely no effect on the outcome in the long run Whether all the deer start out in the forest during year, or if it was 5/5, or whatever, makes no difference in the long run

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