1300 Linear Algebra and Vector Geometry
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1 1300 Linear Algebra and Vector Geometry R. Craigen Office: MH May-June 2017
2 Matrix Inversion Algorithm One payoff from this theorem: It gives us a way to invert matrices. The Matrix Inversion Algorithm To find the inverse of the n n matrix A 1. Augment A with I n to form augmented matrix ( A I ) 2. Use EROs to put this matrix into RREF 3. If the resulting RREF is of the form ( I B ), then A is invertible and B = A 1 4. If the RREF is not of this form, then A is singular EG: Find the inverse of A = : ( A I ) = Express A above as a product of elementary matrices.
3 When matrix inversion fails Let us try inverting A = ( A I ) = R R R 2 R 2 R 1 R 3 R 3 R R 3 R 3 + R R 1 R 1 R At this point we can stop; Ais not invertible. Why? Row of 0s Express the RREF of A, R = in terms of A and elementary matrices: R = E 5 E 4 E 3 E 2 E 1 A A = A A A 1 0 1
4 Why does the Matrix Inversion Algorithm (MIA) work? A simple proof that, if ( A I ) (EROs) ( I B ) then B = A 1 : Recall Theorem says if A is invertible then its RREF is I. So we have E k E 2 E 1 A = I where E i s are elementary matrices Also E k E 2 E 1 ( A I ) = ( I B ) Write U = E k E 2 E 1. Then UA = I. Now U = UI = U(AA 1 ) = (UA)A 1 = IA 1 = A 1 So E k E 2 E 1 = U = A 1. Now ( I B ) = E k E 2 E 1 ( A I ) = A 1 ( A I ) = ( A 1 A A 1 I ) = ( I A 1 ) It follows that B = A 1 as claimed
5 Revisit possible number of solutions of a system Using matrix form Theorem 1.6.1: A system of linear equations has zero, exactly one, or infinitely many solutions Proof: It suffices to prove that, if there are two distinct solutions, then there must be infinitely many. Why? Chalk & Talk Theorem 1.6.2: If n n matrix A is invertible and b is an n 1 column vector then Ax = b has only one solution namely x = A 1 b. Proof: Premultiply the equation by A 1 A 1 (Ax) = (A 1 A)x = I x = x = A 1 b EG: Use = to solve y + z = 1 x + z = 2 x + y = 3
6 An important fact about finding inverses Theorem 1.6.3: Let A and B be square matrices of the same size 1. If BA = I, then A is invertible, and B = A If AB = I, then A is invertible, and B = A 1. Proof: (Part 2) Suppose AB = I. Consider the system Bx = 0. If x 0 is a solution, then Bx 0 = 0. Now, x 0 = I x 0 = (AB)x 0 = A(Bx 0 ) = A0 = 0 Thus x 0 = 0. So the system Bx = 0 has only the trivial solution. By Theorem B is invertible. Postmultiply AB = I by B 1 : ABB 1 = IB 1 So A = B 1. So also B = A 1 Theorem says you needn t check multiplication on both sides to (i) determine whether a matrix (A) is invertible (ii) verify that another matrix (B) is its inverse it suffices to check that either AB = I or BA = I.
7 An extension of Theorem More things equivalent to a matrix being invertible Theorem 1.6.4: Suppose A is an n n matrix. The following statements are equivalent: 1. A is invertible 2. Ax = 0 has only the trivial solution 3. The RREF of A is I n 4. A can be expressed as a product of elementary matrices 5. The system Ax = b is consistent for all columns b 6. The system Ax = b has a unique solution for all columns b Proof: We ll skip we ve basically done it already anyway Recall that a product of invertible matrices is invertible. With Theorem we can show that the converse is also true Theorem Suppose A and B are n n matrices. If AB is invertible then A and B are also invertible. Proof: Chalk & Talk
8 Determining consistency by elimination Here is a simple example of a general class of problems of a type which may arise in applications x + 2y + 3z = a EG: Find all values of a, b, c such that system 3x + 2y + z = b is x + y + z = c consistent. We put in REF (if possible, or something like it): a a a b b 3a c 2a c c 2a b 3a a a c b 3a a a c a + b 4c Consistency 5a + b 4c = 0 a 1 3 linear system in a, b, c Setting b = s, c = t gives general solution (a, b, c) = ( s+4t 5, s, t)
9 Some special kinds of matrices 1.7 Diagonal, triangular, symmetric A square matrix is diagonal if all entries not on the main diagonal are zero EG: ; A square matrix is: ( ) 0 0 ; I 0 1 n and 0 n n (for any n). upper triangular if every entry below the main diagonal is zero; lower triangular if every entry above the main diagonal is zero. ( ) EG: and are upper triangular ( ) and are lower triangular I n and 0 n n are both upper and lower triangular! A matrix that is both upper and lower triangular is diagonal
10 Inverses and powers of diagonal matrices Notice Similarly = = = = Now = I 3 ; so k d 1 0 d1 k 0 In general,... =... for all k = d n 0 d k n and if all d i s are nonzero then this also holds for negative integers k.
11 Multiplying by diagonal matrices ( ) ( ) a Observe = 0 b while In general, ( ) a 0 b 0 = c ( a 2a ) 3a 4b 5b 6b ( a 2b ) 3c 4a 5b 6c To premultiply matrix A by diagonal matrix D, multiply each row of A by the corresponding diagonal entry of D To postmultiply matrix A by diagonal matrix D, multiply each column of A by the corresponding diagonal entry of D Matrix powers and matrix multiplication are much simpler when a diagonal matrix is involved. At the end of this course we shall learn how to take advantage of this fact.
12 Some properties of triangular and diagonal matrices The square matrix A = [a ij ] is upper triangular: all entries to the left of the main diagonal are zero a ij = 0 whenever i > j the ith row starts with at least i 1 zeros (for every i) A is lower triangular: all entries to the right of the main diagonal are zero a ij = 0 whenever i < j the jth column starts with at least j 1 zeros (for every j) Theorem The transpose of an upper triangular matrix is lower triangular The transpose of a lower triangular matrix is upper triangular 2. The product of upper triangular matrices is upper triangular The product of lower triangular matrices is lower triangular 3. A triangular matrix is invertible no diagonal entry is zero 4. The inverse of a triangular matrix is triangular
13 Some important triangular matrices Matrices in RREF are upper triangular Same with matrices in REF The identity matrix is upper and lower triangular ( diagonal) multiply a row by a constant diagonal elementary matrix EG (3 3) Multiply row 2 by 7: E = Elementary matrix for adding one row to another is triangular EG (3 3) Add twice row 2 to row 3: E = EG Subtract three times row 3 from row 1: E = Swapping rows does not give a triangular elementary matrix
14 Arithmetic with symmetric matrices Remember square matrix A is symmetric if A T = A. Theorem 1.7.2: Suppose A and B are symmetric n n matrices and k is a scalar 1. A T is symmetric 2. A ± B are symmetric 3. ka is symmetric That is, the set of symmetric matrices is preserved by (1) taking transpose; (2) adding and subtracting; (3) scalar multiplication. What about matrix multiplication? ( ) ( ) = ( ) ( ) T 7 4 = 12 7 ( ) ( )
15 Some properties of symmetric matrices If A, B are symmetric, then for AB to be symmetric we require AB = (AB) T = B T A T = BA Theorem 1.7.3: The product of symmetric matrices is symmetric if and only if they commute Notice that if A is symmetric then So A 1 is also symmetric (A 1 ) T = (A T ) 1 = A 1 Theorem 1.7.4: If A is invertible and symmetric then so is A 1 Finally, a useful way to manufacture symmetric matrices: If A is any matrix (needn t be square) and B = AA T then B T = (AA T ) T = (A T ) T A T = AA T = B So B = AA T is symmetric. Similarly, C = A T A is symmetric. Theorem 1.7.5: If A is invertible, then so are AA T and A T A.
16 The determinant a matrix function Determinants by cofactor expansion 2.1 A matrix function is a rule that associates some number with every matrix. EG: Recall (from lab) the trace tr(a) of a square matrix A is the sum of its diagonal entries tr = = We now consider an important matrix function for square matrices called the determinant. The determinant of A is denoted det A or (shorthand) A. NOTE: Despite appearances A is not the absolute value!! The rule for det A is simple when A is 1 1 or 2 2: 1 1: det(a) = a (the single scalar entry of a 1 1 matrix) ( ) a b 2 2: det = c d a b c d = ad bc. (Look familiar?)
17 n n determinants for n > 2 For an n n matrix A = [a ij ] we say that the (i, j) minor (or the minor associated with entry a ij M ij ) is the determinant of the (n 1) (n 1) matrix obtained by deleting the ith row and jth column of A. The (i, j) cofactor is defined by C ij = ( 1) i+j M ij EG: Find minors and cofactors for the matrix A = : M 22 = = ( 1) 2+2 M 22 = 12 M 23 = = ( 1) 2+3 M 23 = 6 The checkerboard pattern Easy mnemonic for which positions get which sign (+/ ): (1, 1) position gets any move switches + + a checkerboard pattern results + +
18 The cofactor expansion Also known as Laplace expansion and expansion by minors If we know how to calculate (n 1) (n 1) determinants, we can calculate minors and cofactors of an n n matrix. The cofactor expansion along the ith row of matrix A = [a ij ] n n is the sum a i1 C i1 + a i2 C i2 + + a in C in The cofactor expansion down the jth column of A is the sum a 1j C 1j + a 2j C 2j + + a nj C nj ( ) a b EG: Find all cofactor expansions for generic 2 2 matrix A = c d Row 1: a 11 C 11 + a 12 C 12 = ad + b( c) = ad bc Row 2: a 21 C 21 + a 22 C 22 = a 21 M 21 + a 22 M 22 = cb + da = ad bc Column 1: a 11 M 11 a 21 M 21 = ad cb = ad bc Column 2: a 12 M 12 + a 22 M 22 = bc + da = ad bc
19 A remarkable fact about cofactor expansions Theorem 2.1.1: Every cofactor expansion of an n n matrix, across any row, or down any column, has the same value. The determinant of an n n matrix A is the common value of all of the cofactor expansions (which are properly called cofactor expansions of the determinant of A). Thus, det(a) = A = a i1 C i1 + a i2 C i2 + + a in C in for all i = 1,..., n = a 1j C 1j + a 2j C 2j + + a nj C nj for all j = 1,..., n EG: Find the cofactor expansions and thus the determinant of the a b c 3 3 generic matrix A = d e f along rows 1, 2 and column g h i (Row 1) A = a 11 M 11 a 12 M 12 +a 13 M 13 = a e f h i b d f g i +c d g e h = a(ei fh) b(di fg) + c(dh eg) = aei + bfg + cdh afh bdi cge
20 Basket weaving A shortcut that works only for 3 3 determinants! A quick way to find the determinant of a 3 3 matrix: 1. Adjoin copies of column 1 and column 2 after column 3 2. Add the products of numbers on descending diagonals 3. Subtract the products of numbers on ascending diagonals Thus for our generic 3 3 matrix A we obtain a b c a b c a b A = d e f d e f d e g h i g h i g h aei + bgf + cdh ceg afh bdi EG: Calculate determinants and
21 Smart choices for cofactor expansions Determinants of triangular matrices are a snap! Notice how some expansions of determinants with lots of zeros are less work than others: EG: = Theorem 2.1.2: If A is an n n triangular matrix then det A is the product of the diagonal entries of A Proof: Chalk & talk Examples...
22 A couple more simple properties of determinants We start with some obvious things Theorem 2.2.1: If a square matrix has a row of zeros or a column of zeros, then its determinant is zero Proof: Cofactor expansion along that row or column Theorem 2.2.1: If A is any square matrix then det(a) = det(a T ) Proof: The cofactor expansion along row i of A amounts to the same thing as the cofactor expansion down column i of A T What effect have EROs on determinants? a b c If d e f = x, then: g h i a b c 2d 2e 2f g h i = 2x, d e f a b c g h i = x, a b + 2a c d e + 2d f g h + 2g i = x
23 An even better way to evaluate determinants! How row operations affect the determinant 2.2 Theorem If matrix B is obtained by performing the given EROs on n n matrix A then their determinants are related as follows. 1. Multiply a row or column of A by a scalar k: Then det(b) = k det(a) 2. Interchange two rows or two columns of A: Then det(b) = det(a) 3. Add a multiple of one row of A to another row, or a multiple of one column of A to another column: Then det(b) = det(a) Proof: (1) Expand by minors along that row or column (2) Swap consecutive rows; expand by minors along both rows. Any two rows swapped by an odd number of consecutive swaps. Lemma: If a matrix has two identical rows then its determinant is 0 Proof: Consider the effect of swapping those rows (3) Expand along row being modified. Break the sum into two parts. Interpret these as determinants. Apply the lemma to one of these.
24 May summary ACTION ITEMS and TERMS DEFINED Read 1.7, 2.1, 2.2, start 2.3; preview 3.4; everything in the 1300 Wiki except Linear Transformations and Eigenvalues Do all problems in 1.7, 2.1, 2.2, 2.3 (any material covered) Terms learned: Matrix Inversion Algorithm (MIA); Diagonal, upper triangular and lower triangular matrices; matrix function; trace; determinant; (i, j) minor (minor associated with entry a ij ) M ij ; (i, j) cofactor C ij ; the checkerboard pattern (sign changes for turning minors into cofactors); cofactor expansion along ith row and down jth column ( Laplace expansion or expansion by minors ); the determinant of a matrix (recursively defined in terms of cofactors)
25 May Summary (cont.) KEY CONCEPTS Things equivalent to a matrix being invertible; Checking inverse and invertibility of a square matrix requires only one-sided inverse: BA = I or AB = I (i.e., one-sided) then B = A 1 if a product of square matrices is invertible then the matrices were invertible; matrix arithmetic is easier if one of the matrices is diagonal; upper and lower triangular matrices in terms of which entries a ij must be 0. Properties of triangular matrices; examples of triangular matrices; the product of symmetric matrices is symmetric if and only if they commute; AA T is symmetric in all cases and invertible when A is invertible; determinant is a matrix function; n n determinants will be defined in terms of (n 1) (n 1) determinants; All cofactor expansions of a given matrix are equal; the determinant of a triangular matrix is the product of its diagonal entries; determinants are defined recursively : n n determinants in terms of (n 1) (n 1); determinants of elementary matrices are easy to infer and also the effect of performing EROs on a determinant and the two amount to the same thing; also determinant is unchanged by transpose; a determinant is 0 if a row is 0 or two rows are identical or one is a multiple of the other (same for columns)
26 May Summary (cont.) METHODS LEARNED Proving things are equivalent with a cycle of implications A => B => C => A (etc); How to use the MIA to find the inverse of a matrix or show it is not invertible; expressing an invertible matrix as a product of elementary matrices or expressing the RREF of A (product of elem matr s) A; solving a system in one step when its coefficient matrix is invertible; How to determine when a system (with unknown coefficients) is consistent; how to quickly calculate powers, inverses and multiply (both pre and post) by diagonal matrices; calculating 1 1 and 2 2 determinants; calculating minors and cofactors and cofactor/minor expansions along any row/column of a matrix; finding determinants using cofactor expansion and for 3 3, by basket weaving; strategically picking which cofactor expansion will save work; one-step determinants of triangular matrices; determinants of elementary matrices; effect of EROs on determinants; using these to calculate determinants efficiently by reducing to triangular form; working with determinants in the abstract (i.e. when entries may be unknown or incomplete information is known); inferring values from the effects of operations.
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