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(A, B) = tr(B" A) is an inner product on Cn×n

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 12EQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

To avoid any confusion, unless specified otherwise, vector spaces are complex vector spaces, inner products are complex inner-products, and matrices are complex matrices.

True or False:

(A, B) = tr(BH A) is an inner product
on Cnxn.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

Expert Solution

Step 1

Let us consider the vector space of all $n \times n$ matrices over the field of complex numbers $ℂ$ . Suppose, A and B are two matrices such that $A, B \in M_{n \times n} (ℂ)$ and $<.>$ is defined by defined by

$<A, B> = t r (B^{H} A)$ (i)

To verify whether $<A, B>$ is an inner product space or not

1. For any $A, B, C \in M_{n \times n} (ℂ)$

$\begin{array}{rcl} <A + B, C> & = & t r (C^{H} (A + B)) \\ = & t r (C^{H} A + C^{H} B) [B y p r o p e r t y o f H e r m i t i a n m a t r i x] \\ = & t r (C^{H} A) + t r (C^{H} B) \\ = & <A, C> + <B, C> \end{array}$

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