Let A B denotes the tensor product of matrices Amxn andBpxq defined by the block matrixA B =a12B922Ba11Ba21 B::ami B am2 B:ain Ba2n B:amn Bmpxnq(a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr.Prove that (AB)(C > D) = ACⒸ BD(b) Let Amxm and B₁xn be nonsingular matrices.- Prove that (AB) is nonsingular.- Find the inverse matrix of (AB)(c) Prove that the following equality applies for any Amxmand Bnxn square matricestrace(AB) = trace(A) * trace(B)

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Let A B denotes the tensor product of matrices Amxn and Bpxq defined by the block matrix A & B := a11 B a21 B : am1B a12B a22B ⠀ am2 B a1nB a2n B : amn B mpxng

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.4: Similarity And Diagonalization

Problem 41EQ: In general, it is difficult to show that two matrices are similar. However, if two similar matrices...

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Question

Let A B denotes the tensor product of matrices Amxn and
Bpxq defined by the block matrix
A B =
a12B
922B
a11B
a21 B
:
:
ami B am2 B
:
ain B
a2n B
:
amn B
mpxnq
(a) Let A, B, C, D be defined as Amxn, Bpxq, Cnxk, and Dqxr.
Prove that (AB)(C > D) = ACⒸ BD
(b) Let Amxm and B₁xn be nonsingular matrices.
- Prove that (AB) is nonsingular.
- Find the inverse matrix of (AB)
(c) Prove that the following equality applies for any Amxm
and Bnxn square matrices
trace(AB) = trace(A) * trace(B)

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