By Robert A. Beezer
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Example text
C 2005, 2006 Theorem TMSM Transpose and Matrix Scalar Multiplication Robert A. Beezer 106 Suppose that α ∈ C and A is an m × n matrix. Then (αA)t = αAt . c 2005, 2006 Robert A. Beezer Theorem TT Transpose of a Transpose 107 t Suppose that A is an m × n matrix. Then (At ) = A. c 2005, 2006 Definition CCM Complex Conjugate of a Matrix Robert A. Beezer 108 Suppose A is an m × n matrix. Then the conjugate of A, written A is an m × n matrix defined by A ij = [A]ij c 2005, 2006 Robert A. Beezer Theorem CRMA Conjugation Respects Matrix Addition 109 Suppose that A and B are m × n matrices.
Then the matrix product of A with B is the m × p matrix where column i is the matrix-vector product ABi . Symbolically, AB = A [B1 |B2 |B3 | . . |Bp ] = [AB1 |AB2 |AB3 | . . |ABp ] . c 2005, 2006 Robert A. Beezer Theorem EMP Entries of Matrix Products 121 Suppose A is an m × n matrix and B is an n × p matrix. Then for 1 ≤ i ≤ m, 1 ≤ j ≤ p, the individual entries of AB are given by [AB]ij = [A]i1 [B]1j + [A]i2 [B]2j + [A]i3 [B]3j + · · · + [A]in [B]nj n = [A]ik [B]kj k=1 c 2005, 2006 Theorem MMZM Matrix Multiplication and the Zero Matrix Robert A.
Bp ] = [AB1 |AB2 |AB3 | . . |ABp ] . c 2005, 2006 Robert A. Beezer Theorem EMP Entries of Matrix Products 121 Suppose A is an m × n matrix and B is an n × p matrix. Then for 1 ≤ i ≤ m, 1 ≤ j ≤ p, the individual entries of AB are given by [AB]ij = [A]i1 [B]1j + [A]i2 [B]2j + [A]i3 [B]3j + · · · + [A]in [B]nj n = [A]ik [B]kj k=1 c 2005, 2006 Theorem MMZM Matrix Multiplication and the Zero Matrix Robert A. Beezer 122 Suppose A is an m × n matrix. Then 1. AOn×p = Om×p 2. Op×m A = Op×n c 2005, 2006 Robert A.