Lecture 7: Vectors and Matrices II Introduction to Matrices (See Sections, 3.3, 3.6, 3.7 and 3.9 in Boas)

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1 Lecture 7: Vectors and Matrices II Introduction to Matrices (See Sections and 3.9 in Boas) Here we will continue our discussion of vectors and their transformations. In Lecture 6 we gained some facility with vectors (-tuples) and we now want to discuss generalized operations on vectors a very common activity in physics. In particular we will focus on linear functions of vectors which are themselves sometimes vectors. Consider first a general (scalar valued) function of a vector f r. Recall that a function is a linear function if and only if the following properties hold: f r r2 f r f r2 (7.) f ar af r where a is a scalar (i.e. an ordinary constant). Examples of scalar valued functions of a vector are f r r xˆ x and f2 r r x y z. Comparing to Eq. (7.) we see that f is a linear function and f 2 is not! A similar definition of linearity holds for vector valued functions of vectors as in the following F r r2 F r F r2 (7.2) F ar af r. A trivial example of a linear vector valued function is constant. F r ar multiplication by a ASIDE: he definition of linearity in Eqs. (7.) and (7.2) is the standard definition for a linear operator (e.g. d dt ). In general we have A B A B aa a A. his line of reasoning (linear vector valued functions of vectors) leads us directly to the idea of matrices. As in Lecture 6 we want to connect the idea of vector valued linear functions of general vectors to explicit (concrete) expressions for this operation in terms of components defined by an explicit set of basis vectors v k. k Physics 227 Lecture 7 Autumn 2008

2 Expressing both F and r in terms of components we have (introducing some definitions; note we are assuming that Fr and r are in the same vector space and therefore can be expressed in terms of the same basis vectors) F r F r v R R v F r R r k k k k k k k k r v k k k F r F r v r F v k k k k k k (7.3) where the last step follows due to the linearity of F. Since Fv k is a vector we must be able to express it in terms of its components in the given basis set. Further these components are important and we should give them a name F v F v v M v. k j k j jk j j j (7.4) he x array of numbers F v M j k jk (note the order of the indices) is the matrix mentioned earlier. Substituting back in Eq. (7.3) (and changing the order of the sums) we have F r r F v r M v M r v R v R k k k jk j jk k j j j k k j j k j j k M r. jk k (7.5) he last step follows from the uniqueness of the components which is seen most directly if we choose our basis vectors to be orthogonal (but this is not necessary). We think of M as a square array of numbers where j defines the row and k the jk column of a specific entry. hus given a specific basis set a linear vector valued function of vectors is defined by the corresponding matrix M (just like a vector is jk Physics 227 Lecture 7 2 Autumn 2008

3 defined by its components in the chosen basis). If we think of the set of components of a vector as a x matrix then Eq. (7.5) illustrates matrix multiplication the elements in one row of the left-hand matrix times (one-at-a-time) the elements in a column of the right-hand matrix. Looking back to Lecture 6 (recall Eqs.(6.2) and (6.3)) we see that the scalar product of 2 vectors (in a Euclidean geometry) is just the usual matrix multiplication of an x matrix (the transpose of the vector) times an x matrix r r r r r r 2 2 k 2 k k (7.6) (or in the complex case r r r r r r r r where the dagger means the complex conjugate transpose or Hermitian conjugate). We should also note that Eq. (7.5) has the usual structure of simultaneous linear equations for unknowns in the case we know M jk and the R j and we want to solve for the r k. We will return to this subject in the next Lecture. Clearly matrices are very useful in physics and we want to learn about their properties which will teach us about linear operators simultaneous linear equations and Group heory (see Section 3.3). In the following discussion we will use capital Roman letters to represent linear operators and their abstract matrices. Just as in our discussion of vectors we will be able to define abstract relations independent of any basis vector set and concrete relations that involve the explicit components of vectors and operators (matrices) with respect to specific basis vectors. R Mr abstract R M r M r k l l l concrete where in the very last expression we have employed the (standard) convention of summing over repeated indices. An interesting question is what kind of operations can we apply to the matrices themselves? Clearly we can add (or subtract) matrices (i.e. add linear operators) (7.7) Physics 227 Lecture 7 3 Autumn 2008

4 A B r Ar Br X X A B r k l l A B A B. (7.8) Adding matrices just means adding the corresponding elements. Addition of matrices is associative and commutative A B C A B C A B B A. Also we can multiply matrices by constants (scalars) which is commutative associative and distributive A r Ar X X k l A r l A A A A B A B A A A A (7.9) (7.0) Finally we can define the multiplication of matrices as we did earlier as long as the matrices are conformable i.e. as long as the numbers of rows and columns match. We will generally be considering square matrices x of the same dimension so this point will not be an issue (i.e. our linear operation on a vector produces a vector in the same vector space). We have Physics 227 Lecture 7 4 Autumn 2008

5 AB r A Br X X k lm m lm km A B r AB A B A B l lm lm. (7.) Matrix multiplication is associative distributive but not generally commutative (and this last point is really important in physics) ABC A BC AB C A B C AB AC AB BA except in special cases. (7.2) he difference implied by this last expression AB BA is so important that we give it a name the commutator and a symbol (square brackets) AB BA A B C (7.3) where the commutator has the form of another matrix C. (ote that the commutator only makes sense when A and B are square matrices. Otherwise even if AB and BA both exist they will be of different dimensions.) As we will see the special case of commuting matrices AB BA (AB 0) also plays an important role in physics. (ransformations associated with commuting matrices are called Abelian transformations while non-commuting transformations (matrices) are labeled non- Abelian.) o complete our introduction to matrices we need 2 special matrices (in analogy to our discussion of vectors). he first is the null matrix (all elements zero) 0 0 0r 0 all vectors r (7.4) Physics 227 Lecture 7 5 Autumn 2008

6 and the second is the unit matrix that we met first in Lecture 6 0 r r all vectors r 0 (7.5). ote also that the equality of 2 matrices A B means that Ar Br for all r. In component notation two matrices are equal if and only if every element is identical A B. While still focusing on square matrices a special question (essential in the application of matrices to simultaneous equations) is whether given a matrix A there exists a second matrix B such that AB BA B A (7.6) i.e. we say that B is the inverse matrix to A. Further if exists we say that A is a non-singular matrix (see below). In operator language A represents the inverse of the operation represented by A. For example if A rotates vectors clockwise through some angle about some axis then A rotates them counterclockwise through the same size angle and about the same axis. ASIDE: Even if the matrix A is not square but rectangular 2 2 we can still consider the question of finding an inverse but in this case we must distinguish the left-inverse A A from the right-inverse AA since L AL AR (they necessarily have different dimensions if they exist i.e. they satisfy different conformality constraints and the 2 unit matrices in these equations have different dimensions). his question will not be an issue in this course and we will generally assume that only square matrices can be non-singular. o be able to work with matrices it is useful to define 2 scalar quantities related to each matrix A i.e. these quantities characterize the matrix and are invariant under the usual transformations of the matrix (i.e. scalar means invariant). hey play a role A R Physics 227 Lecture 7 6 Autumn 2008

7 similar to that played by the length of a vector r. he first quantity is the trace of the matrix written as A r A (7.7) i.e. the trace is the sum of the diagonal elements. he second scalar quantity is the determinant which we already assumed in our discussion of the vector product of 2 vectors in Lecture 6. For a 2x2 matrix we have k kk a b a b A det A ad bc. c d c d (7.8) o analyze larger matrices we first define a couple more quantities. For the moment just accept the following definitions. heir intrinsic value will become clear shortly. Consider an x matrix A with elements A. For specific values of the indices k and l construct a new dimensional matrix by eliminating the k th row and the l th column. he determinant of this smaller matrix is called the minor M of the original element A. Since the minor yields a number for each value of k and l these numbers can be thought of as defining another x matrix. ext we multiply the minor by a power of minus defined by and construct the so-called cofactor C of A which is itself an element of a x matrix take away the kth row C M det A. take away the lth column (7.9) For example as applied to a specific 3x3 matrix we have a b c 3 b c A d e f C2 bj ch. h j g h j (7.20) Physics 227 Lecture 7 7 Autumn 2008

8 ow we can use the cofactors to connect 2x2 determinants (which we know from Eq. (7.8)) to the desired x determinants. he general result is that we can evaluate the determinant of an x matrix by first picking a specific (but arbitrary) row or column here we choose row k and then evaluating the sum of each element in that row times its cofactor A A A C k det any fixed (7.2) l where l is summed over and k is not. Since the cofactor itself involves a determinant of a smaller matrix we can iterate this expression in terms of determinants of successively smaller matrices until we reach the 2x2 case (this construction is called the Laplace development for determinants). Square matrices with nonzero determinants are nonsingular matrices and as we will see shortly have inverses. If follows from this definition of the determinant (although we will not go through the arithmetic here) that the determinant of a product of matrices is the product of the individual determinants det[ AB] det[ A] det[ B]. (7.22) Consider some simple examples applying the rules above : A A A A 2 3 2: A A A A A A A A A A A 3: A A A A etc. A A A32 A33 A3 A33 A3 A32 here are several important properties of determinants following from Eq. (7.2) that help to simplify the process of evaluating determinants (recall our motto). ) If (all of the elements in) a single row or a single column are multiplied by a constant c the determinant of the new matrix is c times the determinant of the Physics 227 Lecture 7 8 Autumn 2008

9 original matrix. 2) he sign of a determinant (but not its absolute value) is changed if we interchange 2 rows or 2 columns. 3) he determinant is unchanged if we interchange all the columns with all the rows A A i.e. the determinant of the transpose matrix is the same as the determinant of the original matrix A A A A. 4) he determinant is unchanged if we add a constant times any row to another row or the same for columns. 5) he determinant of a matrix vanishes if all of the elements in any row or column vanish. 6) he determinant of a matrix vanishes if any two rows or any two columns are identical (as follows from points 4) and 5)). 7) he determinant of a matrix vanishes if any two rows or any two columns differ by a constant multiplier i.e. are proportional (as follows from points 4) and 5)). We can use these properties to simplify a matrix i.e. get as many zero elements as possible before taking the determinant (again recall our motto!). As an example consider the following matrix and evaluate its determinant by using the elements in the first row (7.23) Instead consider proceeding by first subtracting 4 times the first row from the second (R2-4*R) then take 7 times the first row from the third row (R3-7*R) and finally subtract 2 times the second row from the third row (R3-2*R2) Physics 227 Lecture 7 9 Autumn 2008

10 R23* R R37* R R32* R (7.24) We obtain the same answer but in some sense the arithmetic is simpler. ote that this analysis also works the other way in the sense that for a vanishing determinant we can always manipulate the form of the determinant by using the rules above into an expression with all zeros along one row or one column. o return to the question of finding the inverse of a matrix consider a slightly edited version of Eq. (7.2) i.e. consider what happens if we multiply a row (or column) of elements by the cofactors of a different row (or column). For example consider the quantity k m A Cml det[ A] (7.25) l which we recognize as yielding the determinant of a matrix A that differs from the matrix A in that both the k th and the m th row of A are equal to the k th row of A. hus these two rows in A are equal to each other and by rule 6) above we can conclude that det A 0. he same is true for summing on the elements of a column. So the right hand side of Eq. (7.25) can be nonzero only if (i.e. we only get a nonzero result for the original expression for the determinant if) k = m A C A C det[ A]. (7.26) ml lm km l l Recalling that the transpose of a matrix is obtained by exchanging rows and columns we are led to consider the product A A (7.27) AC for which we can write using Eq. (7.26) that Physics 227 Lecture 7 0 Autumn 2008

11 AC A C A Cml km det A km lm l l C AC det A A. det A (7.28) Comparing this last equation with Eq. (7.6) we see that we have found a general expression for the inverse of the matrix A. We have A A C det A C det A C det A. (7.29) It should now be clear why nonsingular matrices with inverses require nonzero determinants. he analogue statement to Eq. (7.22) about inverses of products is (note the order of the matrices) AB B A AB AB B A AB B B B B. (7.30) As an explicit example consider a 0 b 2 2 A 0 0 det A aa 0 bb a b b c a a 0 b a bc b C bc a b ac C 0 a b 0 b 0 a b ac a Physics 227 Lecture 7 Autumn 2008

12 a bc b C 2 2 A 0 a b det A a b b ac a (7.3) he reader is encouraged to verify that with the above expressions AA We close this introduction to/review of matrices with a summary of vocabulary/notation and general properties. Operations on matrices: A A. t ranspose A A A A : A A AB B A : AB AB A B B A Complex Conjugate A A : A A Hermitian Conjugate [or Adjoint] A A : A A. ˆ Old Adjoint : ˆ A C A C C lm mk km ml ypes of matrices: Symmetric Matrix A A : A A Antisymmnetric Matrix A A : A A Real Matrix A A : A A Pure Imaginary Matrix A A : A A Orthogonal Matrix : A A A A : AA Hermitian Matrix A A Unitary Matrix : A A A A : AA. : A A Physics 227 Lecture 7 2 Autumn 2008