2005/8Matrices-1 Matrices

2005/8Matrices-2 A Matrix over a Field F (R or C) m rows n columns size: m×n ij-entry: a ij  F (ij-component)

2005/8Matrices-3 The i-row (vector) The j-column (vector) (row matrix) (column matrix) Square matrix: m = n

2005/8Matrices-4 Diagonal matrix Trace unit n  n matrix = diag(1, 1,   , 1) zero matrix = diag(0, 0,   , 0)

2005/8Matrices-5 Example:

2005/8Matrices-6 Equal Example:

2005/8Matrices-7 Matrix addition Example:

2005/8Matrices-8 Matrix Subtraction Scalar Multiplication over a field F (R or C)

2005/8Matrices-9 Matrix Multiplication equal Size of AB

2005/8Matrices-10 Example: Sol:

2005/8Matrices-11 The partitioned matrices submatrix

2005/8Matrices-12 Properties of Matrix Operations Three elementary matrix operations: (1) addition (2) scalar multiplication (3) multiplication zero matrix: identity matrix of order n:

2005/8Matrices-13 The properties of addition and scalar multiplication (1) A + B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A =cA + dA

2005/8Matrices-14 The properties of zero matrix Note ： (1)0 m×n : the addition identity ( 加法單位矩陣 ) (2)  A: the addition inverse ( 加法反元素 ) of matrix A If A  M m  n, and c is a scalar, then (1) A + 0 m  n = A (2) A + (  A) = 0 m  n (3) cA = 0 m  n  c = 0 or A = 0 m  n

2005/8Matrices-15 The properties of matrix multiplication (1) Ａ (BC) = (AB)C (2) Ａ (B+C) = AB + AC (3) (A+B)C = AC + BC (4) c(AB) = (cA) B = A(cB) The properties of identity matrix

2005/8Matrices-16 The transpose ( 轉置 ) of a matrix

2005/8Matrices-17 Ex: Find the transpose of the following matrices. (b)(c) Sol: (a) (b) (c) (a)

2005/8Matrices-18 The properties of transpose matrices

2005/8Matrices-19 The symmetric matrix ( 對稱矩陣 ) If A = A T, then the square matrix A is called symmetric. If A T =  A, then the square matrix A is called skew-symmetric. Example: is symmetric, then find the values of a, b, c. Sol: The skew-symmetric matrix ( 反對稱矩陣 )

2005/8Matrices-20 Ex: is skew-symmetric, find a, b, c. Sol: Note: is symmetric. Pf:

2005/8Matrices-21 Real Numbers ab = baMultiplication commutative Matrices Three possibilities:

2005/8Matrices-22 Ex: For given matrices A and B, show that AB  BA. and Sol:

2005/8Matrices-23 Real numbers ac = bc, Cancellation laws Matrices (1) If C is invertible, then A = B (2) If C is non-invertible, then. (cancellation law does not hold)

2005/8Matrices-24 Ex ： For given matrices A, B and C, show that AC=BC. Sol:, but.