Matrix representation of a linear map

Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors (of size m) viz., f (e 1 ),..., f (e n ). Place them side by side., Thus if f (e j ) = (f 1j, f 2j,..., f mj ) T, then we obtain an array of the form M f = f 11 f 12... f 1n f 21 f 22... f 2n.. f m1 f m2... f mn. Such an array is called a matrix with m rows and n columns or a matrix of size m n. In short, we write M f = (f ij ). We call M f to be the matrix associated with the linear map f : R n R m. 12/45

13/45 If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m).

If we let Mat m n (R), or simply, M m,n, denote the set of all m n matrices with entries in R, then we obtain: M : L(n, m) M m,n f M f The map M is a one-to-one correspondence (i.e., one-one and onto map) between the set of all linear maps from R n to R m and the set of all m n matrices with entries in R. Note: Two matrices are equal if their sizes are the same and corresponding entries are the same. M is one-one (or injective), means M f = M g for some f, g L(n, m) f = g, whereas M is onto (or surjective) means that A Mat m n (R) A = M f for some f L(n, m). M preserves addition and scalar multiplication, i.e., M f +g = M f +M g and M αf = αm f, f, g L(n, m), α R. 13/45

Examples: 14/45

14/45 Examples: 1. For the identity map id : R n R n, 1 0... 0 0 1... 0 M id = I = I n :=.. 0 0... 1

14/45 Examples: 1. For the identity map id : R n R n, 1 0... 0 0 1... 0 M id = I = I n :=.. = (δ ij) 0 0... 1 wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j.

14/45 Examples: 1. For the identity map id : R n R n, 1 0... 0 0 1... 0 M id = I = I n :=.. = (δ ij) 0 0... 1 wehre δ ij is the Kronecker delta defined by δ ij = 1 if i = j and δ ij = 0, if i j. 2. The matrix of the linear map T : R 2 ( R 2 which ) 0 1 interchanges coordinates, is M T =. 1 0 3. For the linear map µ α : R n R n corresponding to multiplication by a fixed scalar α, the associated matrix is

15/45 4. Rotation of the plane by an angle θ defines a linear map R θ : R 2 R 2 (Linearity of R θ can be shown by the law of congruent triangles or by finding a formula for R θ (x, y) using polar coordinates and trigonometric identities). The matrix of R θ is about the ( ) cos θ sin θ M Rθ = sin θ cos θ 5. The inclusion map R 2 R 3 that sends (x, y) R 2 to (x, y, 0) R 3 is clearly a linear map and its matrix is

Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n

Operations on matrices 16/45 Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ). Transpose operation introduced earlier can be extended to all matrices A T := (b ij ) M n,m where b ij := a ji Note that (αa + βb) T = αa T + βb T for any α, β R.

Operations on matrices Set of all matrices of size m n is denoted by M m,n As before, for A = (a ij ), B = (b ij ) M m,n and α R, A + B := (a ij + b ij ); αa := (αa ij ) Matrix addition and scalar multiplicaion satisfies the usual properties. The zero element is the zero matrix 0, all of whose entries are 0 and the negative of a matrix A = (a ij ) is the matrix A = ( a ij ). Transpose operation introduced earlier can be extended to all matrices A T := (b ij ) M n,m where b ij := a ji Note that (αa + βb) T = αa T + βb T for any α, β R. Products of linear maps is not linear (in fact not even defined), in general. But composites of linear maps are linear. How does this relate to matrices? 16/45

17/45 Composition of Linear Maps and Matrix Multiplication Question: Let f : R n R m, g : R p R n be linear maps. If A := M f and B := M g then M f g =?

Composition of Linear Maps and Matrix Multiplication Question: Let f : R n R m, g : R p R n be linear maps. If A := M f and B := M g then M f g =? f g is a map of R p to R m. For 1 j p, ( n ) (f g)(e j ) = f (g(e j )) = f b kj e k = = n b kj f (e k ) = k=1 k=1 ( n m ) b kj a ik e i k=1 ( m n ) a ik b kj e i i=1 k=1 Thus M f g = C, where C = (c ij ) is the m p matrix whose (i, j) th entry is c ij = n k=1 a ikb kj. We call C the product of the m n matrix A and the n p matrix B, and write C = AB. i=1 17/45

Basic Properties of Matrix Multiplication 18/45 The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C.

Basic Properties of Matrix Multiplication The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C. 2. Right and Left Distributivity: A(B + C) = AB + AC, (B + C)A = BA + CA 3. Multiplicative identity: For any A M m,n, AI n = A and I m A = A In particular, if A is a n n matrix, then AI n = A = I n A. Remark: Matrix multiplication is not commutative in general, i.e., AB need not equal BA even when both the products are defined. Exercise: Find examples! 18/45

Basic Properties of Matrix Multiplication The following properties are readily derived from the corresponding properties of linear maps and the one-to -one correspondence between linear maps and matrices. Assume that the matrices A, B, C are of appropriate sizes so that all relevant products are defined. 1. Associativity: A(BC) = (AB)C. 2. Right and Left Distributivity: A(B + C) = AB + AC, (B + C)A = BA + CA 3. Multiplicative identity: For any A M m,n, AI n = A and I m A = A In particular, if A is a n n matrix, then AI n = A = I n A. 4. Transpose of Product: (AB) T = B T A T. Remark: Matrix multiplication is not commutative in general, i.e., AB need not equal BA even when both the products are defined. Exercise: Find examples! 18/45

Invertible Maps 19/45 Recall the set-theoretic definition of invertible functions. Let X, Y be sets. A function f : X Y is said to be invertible if there exists g : Y X satisfying g f = id X and f g = id Y. In this case the function g is unique (check!) and it is called the inverse of f and denoted by f 1.

Invertible Maps Recall the set-theoretic definition of invertible functions. Let X, Y be sets. A function f : X Y is said to be invertible if there exists g : Y X satisfying g f = id X and f g = id Y. In this case the function g is unique (check!) and it is called the inverse of f and denoted by f 1. For any function f : X Y, f is invertible f is one-one and onto. If f : R n R n is an invertible linear map, then f 1 is linear. [To see this, let y 1, y 2 R n. Then there are x 1, x 2 R n such that f (x i ) = y i, i.e., x i = f 1 (y i ) for i = 1, 2. Thus f 1 (y 1 + y 2 ) = f 1 (f (x 1 ) + f (x 2 )) and using the linearity of f, we find that this is equal to f 1 (f (x 1 + x 2 )) = x 1 + x 2 = f 1 (y 1 ) + f 1 (y 2 ). Similarly f 1 preserves scalar multiplication. 19/45

Invertible Matrices 20/45 A square matrix A of size n n is said to be invertible if there exists an n n matrix B such that AB = BA = I n. If such a matrix B exists, then it is unique.

Invertible Matrices 20/45 A square matrix A of size n n is said to be invertible if there exists an n n matrix B such that AB = BA = I n. If such a matrix B exists, then it is unique. Indeed, if C is another matrix such that CA = AC = I n, then C = CI n = C(AB) = (CA)B = I n B = B. We thus call B to be the inverse of A and denote it by A 1. Note that if f A, f B : R n R n are the linear maps associated with A, B resp., then B = A 1 implies that f B = f 1 A.

Invertible Matrices A square matrix A of size n n is said to be invertible if there exists an n n matrix B such that AB = BA = I n. If such a matrix B exists, then it is unique. Indeed, if C is another matrix such that CA = AC = I n, then C = CI n = C(AB) = (CA)B = I n B = B. We thus call B to be the inverse of A and denote it by A 1. Note that if f A, f B : R n R n are the linear maps associated with A, B resp., then B = A 1 implies that f B = f 1 A. Basic Properties/Examples: If A 1, A 2 are invertible, then so is A 1 A 2. What is its inverse? Ans: (A 1 A 2 ) 1 = A 1 2 A 1 1. Clearly I n and more generally, any diagonal matrix diag(a 1, a 2,..., a n ) with a i 0 for all i is invertible; diag(a 1,..., a n ) 1 = diag(1/a 1,..., 1/a n ). 20/45

Elementary Matrices 21/45 Consider a m m matrix E ij whose (i, j) th entry is 1 and all other entries are 0.

Elementary Matrices 21/45 Consider a m m matrix E ij whose (i, j) th entry is 1 and all other entries are 0. If we multiply a m n matrix A by E ij on the left, then what we get is a m n matrix whose i th row is equal to the j th row of A and all other rows are zero. In particular, if i j, then E ij E ij = 0 and so for any α, (I + αe ij )(I αe ij ) = I + αe ij αe ij α 2 E ij E ij = I.

Elementary Matrices Consider a m m matrix E ij whose (i, j) th entry is 1 and all other entries are 0. If we multiply a m n matrix A by E ij on the left, then what we get is a m n matrix whose i th row is equal to the j th row of A and all other rows are zero. In particular, if i j, then E ij E ij = 0 and so for any α, (I + αe ij )(I αe ij ) = I + αe ij αe ij α 2 E ij E ij = I. On the other hand, if i = j, then (I + αe ii ) = diag(1,..., 1, 1 + α, 1,..., 1) and this is invertible if α 1. Matrices of the type I + αe ij, with α R and i j or of the type I + αe ii with α 1 provide simple examples of invertible matrices whose inverse is of a similar type. These are two among 3 possible types of elementary matrices. 21/45

The third remaining type of elementary matrix and its basic propery is described in the following easy exercise. 22/45 Exercise: Given any i j, show that the square matrix T ij := I + E ij + E ji E ii E jj is precisely the matrix obtained from the identity matrix by interchanging the i th and j th rows. Also show that T ij T ij = I and deduce that T ij is invertible and T 1 ij = T ij. A square matrix is said to be elementary if it of the type T ij (i j) or I + αe ii (α 1) or I + αe ij (i j). Thanks to the discussion in the last slide and the exercise above, we have the following result.