Chapter 1.2 MatrixAlgebra

Trace and transpose outline matrices matrix algebra

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Unformatted text preview: s a matrix equation. Trace and Transpose Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose The Trace of a Matrix Definition If A = [aij ] is an n × n matrix, then the trace of A is the sum tr(A) of the n elements along the main diagonal of A; i.e., tr(A) = a11 + a22 + · · · + ann . Example Find the trace of the matrix: 1 2 3 4 −1 −3 −4 −2 A= 1 −1 0 1 0 2 3 4 Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose The Transpose of a Matrix Theorem Let A and B be n × m matrices and let k be a scalar. Then whenever the operations are defined: (a) If A = [aij ], then AT = [bij ], where bij = aji . (b) (AT )T = A. (c) (A + B )T = AT + B T . (d) (kA)T = kAT . (e) (AB )T = B T AT . Outline Matrices Matrix Algebra Matrix Equations Trace and Transpose The Transpose of a Matrix Theorem Let A and B be n × m matrices and let k be a scalar. Then whenever the operations are defined: (a) If A = [aij ], then AT = [bij ], where bij = aji . (b) (AT )T = A. (c) (A + B )T = AT + B T . (d) (kA)T = kAT . (e) (AB )T = B T AT . Outline Matrices Matrix Algebra Matrix Equations Examples Example 1 Find the transpose and trace of the matrix: 1 2 3 4 −1 −3 −4 −2 A= 1 −1 0 1 0 2 3 4 2 Find the transpose of the matrix: B= 1 −3 2 5 −4 −7 Trace and Transpose...
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