function Matrices Calculus III Summer 2013, Session II Wednesday, July 10, 2013
Agenda function 1. 2. function
function Definition An m n matrix is a rectangular array of numbers arranged in m horizontal rows and n vertical columns. These numbers are called the entries or elements of the matrix. Example a 11 a 12... a 1n a 21 a 22... a 2n A =...... a m1 a m2... a mn is an m n matrix. It can be written more succinctly as A = [a ij ]. Two matrices are equal if they have the same size (identical numbers of rows and columns) and the same entries.
Row and column vectors function Definition A 1 n matrix is called a row n-vector, or simply a row vector. An n 1 matrix is called a column n-vector, or a column vector. The elements of a such a vector are its components. Examples 1. The matrix a = [ 2 3 1 4 5 7] is a row 3-vector. 1 2. b = 1 3 is a column 4-vector. 4
function Row and column vectors Any matrix can be written as a list of row or column vectors. Example The matrix has three row 4-vectors: and we can write 2 1 3 4 A = 1 2 1 1 3 1 2 5 a 1 = [ 2 1 3 4 ], a 2 = [ 1 2 1 1 ], and a 3 = [ 3 1 2 5 ] a 1 A = a 2. a 3
function Row and column vectors Any matrix can be written as a list of row or column vectors. Example The matrix 2 1 3 4 A = 1 2 1 1 3 1 2 5 has four column 3-vectors: 2 1 3 4 b 1 = 1, b 2 = 2, b 3 = 1, and b 4 = 1 3 1 2 5 and we can write A = [ b 1 b 2 b 3 b 4 ].
Transpose function Definition If A is the matrix A = [a ij ], the transpose of A is the matrix A T = [a ji ]. If A is an m n matrix then A T is an n m matrix. Example Suppose A is the matrix Then A = [ ] 1 2 3. 4 5 6 1 4 A T = 2 5. 3 6
Types of matrices function Square An n n matrix is called a square matrix since it has the same number of rows and columns. The elements a ii make up the main diagonal. Triangular A square matrix is called upper triangular if a ij = 0 whenever i > j, that is, it has only zeros below the main diagonal. A lower triangular matrix is a square matrix with only zeros above the main diagonal, that is, a ij = 0 whenever i < j. Diagonal A diagonal matrix is a square matrix whose only nonzero entries lie along the main diagonal, that is, a ij = 0 whenever i j.
Types of matrices function Symmetric A matrix satisfying A T = A is called a symmetric matrix. Skew-symmetric A matrix that satisfies A T = A is called skew-symmetric. Notice that both symmetric and skew-symmetric matrices must be square (because if A is m n then A T is n m), a skew-symmetric matrix must have zeros along its main diagonal (because a ii = a ii ).
functions function Definition A matrix function is like a matrix, but replaces numbers with functions of a single real variable. Column vector functions and row vector functions are analogously defined. Example A(t) is a 2 3 matrix function: [ t 3 5 ] t cos t A(t) = t e t2 ln(t + 1) te t. The matrix function is only defined for values of t such that all elements are defined. In this example, A(t) is defined for values of t such that t 0 and t + 1 > 0.
function Definition If A = [a ij ] and B = [b ij ] are matrices with the same dimensions, their sum is A + B = [a ij + b ij ]. addition Similarly, their difference is Example We have [2 ] 1 3 + 4 5 0 and [ ] 2 1 3 4 5 0 A B = [a ij b ij ]. [ ] 1 0 5 = 5 2 7 [ ] 1 0 5 = 5 2 7 [ 1 1 ] 8 1 3 7 [ ] 3 1 2. 9 7 7
Scalar multiplication function A scalar is a real or complex number, as opposed to a vector or matrix. Definition If A is a matrix and s a scalar, then the product of s with A is the matrix obtained by multiplying every element of A by s. Symbolically, if A = [a ij ] then sa = [sa ij ]. Examples [ ] 2 1 If A = then 5A = 4 6 [ ] 10 5. 20 30 If A and B are matrices with the same dimensions then A B = A + ( 1)B.
Matrices behave like you expect function addition, subtraction, and scalar multiplication have familiar properties: A + B = B + A A + (B + C) = (A + B) + C 1A = A s(a + B) = sa + sb (s + t)a = sa + ta s(ta) = (st)a = (ts)a = t(sa) 0... 0 A + 0 = A 0 =..... A A = 0 0... 0 0A = 0... but matrix multiplication does not!
multiplication function Example P = C = sprockets [ belts propellers ] widget A 5 2 3 P C = widget B 6 5 0 supplier 1 supplier 2 sprockets 0.10 0.15 belts 0.50 0.30 propellers 1.00 1.10 supplier 1 [ supplier 2 ] widget A 4.50 4.65 widget B 3.10 2.40
function multiplication Definition Let A = [a ij ] be an m n matrix and B = [b jk ] be an n p matrix. Their product is the m p matrix n AB = [c ik ] where c ik = a ij b jk. j=1 If write write A as a matrix of rows and B as a matrix of columns, a 1 A =. a m and B = [ b 1 b p ], then we can express their product using the vector dot product AB = [a i b k ].
function Familiar properties of matrix multiplication In most ways matrix multiplication behaves like multiplication of scalars: A(BC) = (AB)C A(B + C) = AB + AC (A + B)C = AC + BC (sa)b = s(ab) = A(sB) Definition The identity matrix, I n (or just I), is the n n diagonal matrix with ones on the main diagonal. I 2 = [ ] 1 0, I 0 1 3 = If A is an m n matrix then 1 0 0 0 1 0, etc. 0 0 1 AI n = A and I m A = A.
multiplication is not commutative function If A and B are n n matrices, it is not always true that AB = BA. Example [ ] [ ] 1 2 3 1 If A = and B = then 1 3 2 1 [ ] [ ] 1 2 3 1 AB = = 1 3 2 1 but [ ] [ ] 3 1 1 2 BA = = 2 1 1 3 [ ] 7 1 3 4 [ ] 2 9. 3 1
function function All of the operations we discussed can be applied to matrix functions. In the case of scalar multiplication, a matrix function can be multiplied by any scalar function. Example [ ] 2 + t e If s(t) = e t 2t and A(t) =, their product is 4 cos t [ e s(t)a(t) = t ( 2 + t) e 3t ] 4e t e t. cos t
function function Additionally, we can do calculus with matrix functions! Definition Suppose A(t) = [a ij (t)] is a matrix function. Its derivative is [ ] da dt = daij (t) dt and its integral over the interval [a, b] is b [ b ] A(t) dt = a ij (t) dt. Theorem ( product rule) a If A and B are differentiable matrix functions and the product AB is defined then d (AB) = AdB dt dt + da dt B. a
function function Example [ ] 2t 1 Let A(t) = 6t 2 4e 2t. We have da dt = [ 2 0 12t 8e 2t ] and 1 A(t) dt = 0 [ ] 1 1 2 2e 2. 2