Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q22E (page 413)

If the two columns $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ of a $2 脳 2$ matrix $A$ are orthogonal, then the singular values of $A$ must be $| |\overset{鈫�}{v}| |$ and $| |\overset{鈫�}{w}| |$ .

Short Answer

Expert verified

The given statement is TRUE.

Step by step solution

Definition of singular values:

The singular values of $n 脳 m$ matrix $A$ are the square roots of the eigen values of the symmetric $m 脳 m$ matrix $A^{T} A$ .

To Find TRUE or FALSE

Find $A^{T} A$ as follows:

role="math" localid="1664176579719" $A^{T} A = [\begin{matrix} {\overset{藱}{v}}^{T} \overset{藱}{v} & {\overset{藱}{v}}^{T} \overset{藱}{w} \\ {\overset{藱}{w}}^{T} \overset{藱}{v} & {\overset{藱}{w}}^{T} \overset{藱}{w} \end{matrix}] A^{T} A = [\begin{matrix} {|\overset{藱}{|v|}|}^{2} & 0 \\ 0 & {|\overset{藱}{|w|}|}^{2} \end{matrix}]$

This shows that the eigenvalues of $A^{T} A$ are ${|\overset{藱}{|v|}|}^{2}$ and ${|\overset{藱}{|w|}|}^{2}$ . Hence, it follows that the singular values of $A$ are $|\overset{藱}{|v|}|$ and $|\overset{藱}{|w|}|$ .

Thus, the given statement isTRUE.

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91影视

If the two columns $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ of a $2 脳 2$ matrix $A$ are orthogonal, then the singular values of $A$ must be $| |\overset{鈫�}{v}| |$ and $| |\overset{鈫�}{w}| |$ .

Short Answer

Step by step solution

Definition of singular values:

To Find TRUE or FALSE

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91影视

If the two columns v鈫�andw鈫�of a2脳2matrixAare orthogonal, then the singular values ofAmust be|v鈫�| and|w鈫�|.

Short Answer

Step by step solution

Definition of singular values:

To Find TRUE or FALSE

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Pure Maths

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

If the two columns $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ of a $2 脳 2$ matrix $A$ are orthogonal, then the singular values of $A$ must be $| |\overset{鈫�}{v}| |$ and $| |\overset{鈫�}{w}| |$ .