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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q39E (page 414)

39. The equation $A^{T} A = {AA}^{T}$ holds for all square matrices.

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Check whether the given statement is TRUE or FALSE

Let

$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Then, the transpose of A is:

role="math" localid="1659675610776" $A' = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}]$

Now, find ${AA}^{T} and A^{T} A$ as follows:

${AA}^{T} = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}] = [\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}] A^{T} A = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}]$

Here, ${AA}^{T} = A^{T} A$ .

02

Final Answer

The given statement is FALSE.

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Most popular questions from this chapter

If A is an invertible $2 x 2$ matrix, what is the relationship between the singular values of A and $A^{- 1}$ ? Justify your answer in terms of the image of the unit circle.

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61.qis indefinite and det A < 0

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$q (\overset{鈬赌}{x}) = \overset{鈬赌}{x} . A \overset{鈬赌}{x}$

where A is a symmetricnxnmatrix. Let $\overset{鈬赌}{v}$ be a unit eigenvector of A, with associated eigenvalue $位$ . Find $q (\overset{鈬赌}{v})$ .

Determine the definiteness of the quadratic forms in Exercises 4 through 7.

4. $q (x_{1}, x_{2}) = 6 x_{1}^{2} + 4 x_{1} x_{2} = 3 x_{2}^{2}$

If A is a positive definite $n 脳 n$ matrix, and R is any real $n 脳 n$ matrix, what can you say about the definiteness of the matrix $R^{T} AR$ ? For which matrices R is $R^{T} AR$ positive definite?

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