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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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.
 Spring '14
 KenyonJ.Platt
 Linear Algebra, Algebra, Equations, Matrices

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