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Chapter 7: Q7.4-18E (page 395)

In Exercises 17鈥�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 \(m \times n\) 鈥渄iagonal鈥� matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

18. Suppose \(A\) is square and invertible. Find a singular value decomposition of \({A^{ - 1}}\)

Short Answer

Expert verified

The required singular decomposition of \({A^{ - 1}}\) is \(V\mathop \Sigma \limits^{ - 1} {U^T}\).

Step by step solution

01

Find \(A\)

It is given that\(A\)is square and invertible. As the matrix\(V\)is an orthogonal matrix,\({V^T} = {V^{ - 1}}\), then we have,

\(\begin{array}{c}A = U\Sigma {V^T}\\ = U\Sigma {V^{ - 1}}\end{array}\)

02

Find the singular value decomposition of the matrix \({A^{ - 1}}\)

\(\begin{array}{c}{A^{ - 1}} = {\left( {U\Sigma {V^{ - 1}}} \right)^{ - 1}}\\ = V\mathop \Sigma \limits^{ - 1} {U^{ - 1}}\\ = V\mathop \Sigma \limits^{ - 1} {U^T}\end{array}\)

Therefore, \({A^{ - 1}} = V\mathop \Sigma \limits^{ - 1} {U^T}\).

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Most popular questions from this chapter

25.Let \({\bf{T:}}{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{m}}}\) be a linear transformation. Describe how to find a basis \(B\) for \({\mathbb{R}^n}\) and a basis \(C\) for \({\mathbb{R}^m}\) such that the matrix for \(T\) relative to \(B\) and \(C\) is an \(m \times n\) 鈥渄iagonal鈥� matrix.

Compute the quadratic form \({x^T}Ax\), when \(A = \left( {\begin{aligned}{{}}3&2&0\\2&2&1\\0&1&0\end{aligned}} \right)\) and

a. \(x = \left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right)\)

b. \(x = \left( {\begin{aligned}{{}}{ - 2}\\{ - 1}\\5\end{aligned}} \right)\)

c. \(x = \left( {\begin{aligned}{{}}{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\\{\frac{1}{{\sqrt 2 }}}\end{aligned}} \right)\)

Let \(A = PD{P^{ - {\bf{1}}}}\), where P is orthogonal and D is diagonal, and let \(\lambda \) be an eigenvalue of A of multiplicity k. Then \(\lambda \) appears k times on the diagonal of D.Explain why the dimension of the eigenspace for \(\lambda \) is k.

Classify the quadratic forms in Exercises 9鈥�18. Then make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct \(P\) using the methods of Section 7.1.

12.\({\bf{ - }}x_{\bf{1}}^{\bf{2}}{\bf{ - 2}}{x_{\bf{1}}}{x_{\bf{2}}} - x_{\bf{2}}^{\bf{2}}\)

Question: In Exercises 15 and 16, construct the pseudo-inverse of \(A\). Begin by using a matrix program to produce the SVD of \(A\), or, if that is not available, begin with an orthogonal diagonalization of \({A^T}A\). Use the pseudo-inverse to solve \(A{\rm{x}} = {\rm{b}}\), for \({\rm{b}} = \left( {6, - 1, - 4,6} \right)\) and let \(\mathop {\rm{x}}\limits^\^ \)be the solution. Make a calculation to verify that \(\mathop {\rm{x}}\limits^\^ \) is in Row \(A\). Find a nonzero vector \({\rm{u}}\) in Nul\(A\), and verify that \(\left\| {\mathop {\rm{x}}\limits^\^ } \right\| < \left\| {\mathop {\rm{x}}\limits^\^ + {\rm{u}}} \right\|\), which must be true by Exercise 13(c).

16. \(A = \left( {\begin{array}{*{20}{c}}4&0&{ - 1}&{ - 2}&0\\{ - 5}&0&3&5&0\\{\,\,\,2}&{\,\,0}&{ - 1}&{ - 2}&0\\{\,\,\,6}&{\,\,0}&{ - 3}&{ - 6}&0\end{array}} \right)\)
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