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Chapter 7: Q7.4-20E (page 395)

In Exercises 17–24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) “diagonal” matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

20. Show that if\(A\)is an orthogonal\(m \times m\)matrix, then \(PA\) has the same singular values as \(A\).

Short Answer

Expert verified

It is verified that \(P\) is an orthogonal square matrix and, \(A\) and \(PA\) have the same singular values.

Step by step solution

01

Show that \(PU\) is orthogonal

Consider the equation as:

\(\begin{array}{c}PA = P\left( {U\Sigma {V^T}} \right)\\ = \left( {PU} \right)\Sigma {V^T}\end{array}\)

Since \(P\) and \(U\) are orthogonal, \(PU\) is also orthogonal.

02

Show that the matrices \(A\) and \(PA\) have the same singular values

Since \(PU\)and\(V\)are orthogonal and\(\Sigma \) is a diagonal matrix, so that \(PA = \left( {PU} \right)\Sigma {V^T}\) is the singular value decomposition of\(PA\).

Therefore, the diagonal entries of\(\Sigma \)are the singular values of\(PA\). Hence,

The matrices \(A\) and \(PA\) have the same singular values.

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Most popular questions from this chapter

Question: 4. Let A be an \(n \times n\) symmetric matrix.

a. Show that \({({\rm{Col}}A)^ \bot } = {\rm{Nul}}A\). (Hint: See Section 6.1.)

b. Show that each y in \({\mathbb{R}^n}\) can be written in the form \(y = \hat y + z\), with \(\hat y\) in \({\rm{Col}}A\) and z in \({\rm{Nul}}A\).

Question: In Exercises 1 and 2, convert the matrix of observations to mean deviation form, and construct the sample covariance matrix.

\(1.\,\,\left( {\begin{array}{*{20}{c}}{19}&{22}&6&3&2&{20}\\{12}&6&9&{15}&{13}&5\end{array}} \right)\)

Suppose\(A = PR{P^{ - {\bf{1}}}}\), where P is orthogonal and R is upper triangular. Show that if A is symmetric, then R is symmetric and hence is actually a diagonal matrix.

Construct a spectral decomposition of A from Example 3.

Suppose Aand B are orthogonally diagonalizable and \(AB = BA\). Explain why \(AB\) is also orthogonally diagonalizable.
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