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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: Compute the singular values of the \({\bf{4 \times 4}}\) matrix in Exercise 9 in Section 2.3, and compute the condition number \(\frac{{{\sigma _1}}}{{{\sigma _4}}}\).

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix\(P\)and a diagonal matrix\(D\). To save you time, the eigenvalues in Exercises 17–22 are: (17)\( - {\bf{4}}\), 4, 7; (18)\( - {\bf{3}}\),\( - {\bf{6}}\), 9; (19)\( - {\bf{2}}\), 7; (20)\( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

14. \(\left( {\begin{aligned}{{}}{\,1}&{ - 5}\\{ - 5}&{\,\,1}\end{aligned}} \right)\)

Question: If A is \(m \times n\), then the matrix \(G = {A^T}A\) is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 9 and 10).

10. Show that if an \(n \times n\) matrix G is positive semidefinite and has rank r, then G is the Gram matrix of some \(r \times n\) matrix A. This is called a rank-revealing factorization of G. (Hint: Consider the spectral decomposition of G, and first write G as \(B{B^T}\) for an \(n \times r\) matrix B.)

Let \(A = \left( {\begin{aligned}{{}}{\,\,\,4}&{ - 1}&{ - 1}\\{ - 1}&{\,\,\,4}&{ - 1}\\{ - 1}&{ - 1}&{\,\,\,4}\end{aligned}} \right)\), and\({\rm{v}} = \left( {\begin{aligned}{{}}1\\1\\1\end{aligned}} \right)\). Verify that 5 is an eigenvalue of \(A\) and \({\rm{v}}\)is an eigenvector. Then orthogonally diagonalize \(A\).

Orhtogonally diagonalize the matrices in Exercises 37-40. To practice the methods of this section, do not use an eigenvector routine from your matrix program. Instead, use the program to find the eigenvalues, and for each eigenvalue \(\lambda \), find an orthogonal basis for \({\bf{Nul}}\left( {A - \lambda I} \right)\), as in Examples 2 and 3.

37. \(\left( {\begin{aligned}{{}}{\bf{6}}&{\bf{2}}&{\bf{9}}&{ - {\bf{6}}}\\{\bf{2}}&{\bf{6}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{9}}&{ - {\bf{6}}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{\bf{9}}&{\bf{2}}&{\bf{6}}\end{aligned}} \right)\)

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