Q33E If the singular values of an n脳... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q33E (page 413)

If the singular values of an $n 脳 n$ matrix A are all 1 , A is necessarily orthogonal?

Short Answer

Expert verified

Use the fact that $A^{t} A$ is diagonalizable

Step by step solution

To finding A necessarily orthogonal for value of an matrix A

Let A be an $n 脳 n$ matrix such that the singular values of A are all 1. This implies that, all the eigenvalues of the matrix $A^{t} A$ are 1.

Note that, $A^{t} A$ is a symmetric matrix since ${(A^{t} A)}^{t} = A^{t} {(A^{t})}^{t} = A^{t} A .$ Hence $A^{t} A$ is diagonalizable. Thus, there exists a diagonal matrixand an invertible matrix P such that

$A^{t} A = P D P^{- 1}$

To finding A necessarily orthogonal for value of an matrix A

Also, from the construction of D , we know that the diagonal entries of D are basically the eigenvalues of $A^{t} A$ .

Here, all the eigenvalues of the matrix $A^{t} A$ are 1. This shows that all the diagonal entries of D are 1. Hence $D = I_{n}$ . Using this we get that,

$A^{t} A = P D P^{- 1} = P I_{n} P^{- 1} = P P^{- 1} = I_{n}$

Hence $A^{t} A = I_{n}$ which implies that A is orthogonal.

Step 3:Final proof

Use the fact that $A^{t} A$ is diagonalizable.

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91影视

If the singular values of an $n 脳 n$ matrix A are all 1 , A is necessarily orthogonal?

Short Answer

Step by step solution

To finding A necessarily orthogonal for value of an matrix A

To finding A necessarily orthogonal for value of an matrix A

Step 3:Final proof

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91影视

If the singular values of an n脳nmatrix A are all 1 , A is necessarily orthogonal?

Short Answer

Step by step solution

To finding A necessarily orthogonal for value of an matrix A

To finding A necessarily orthogonal for value of an matrix A

Step 3:Final proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

If the singular values of an $n 脳 n$ matrix A are all 1 , A is necessarily orthogonal?