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Chapter 7: Q7.4-19E (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.

19. Show that the columns of\(V\)are eigenvectors of\({A^T}A\), the columns of\(U\)are eigenvectors of\(A{A^T}\), and the diagonal entries of\({\bf{\Sigma }}\)are the singular values of \(A\). (Hint: Use the SVD to compute \({A^T}A\) and \(A{A^T}\).)

Short Answer

Expert verified

It is verified that, \(V\) and \(U\) are eigenvectors of \({A^T}A\) and \(A{A^T}\) respectively, and the columns of \(U\) re eigenvectors of \(A{A^T}\) and the diagonal elements of \(\Sigma \) are the singular values of \(A\).

Step by step solution

01

Find the product of \({A^T}\) and \(A\)

As singular value decomposition of\(A\)is \(A = U\Sigma {V^T}\), then

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

Therefore, from the Diagonalization theorem. Singular values are the diagonal entries in \(\Sigma \) matrix. The columns of matrix \(V\) are the eigenvectors of \({A^T}A\).

02

Find the product of \(A\) and \({A^T}\)

Now, find the product of \(A = U\Sigma {V^T}\) and its transpose.

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

Thus, it can be observed that\(U\)diagonalizes\(A{A^T}\)and the columns of\(U\)must be eigenvectors of\(A{A^T}\).

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Most popular questions from this chapter

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

(M) Compute an SVD of each matrix in Exercises 26 and 27. Report the final matrix entries accurate to two decimal places. Use the method of Examples 3 and 4.

26. \(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{ - {\bf{18}}}&{{\bf{13}}}&{ - {\bf{4}}}&{\bf{4}}\\{\bf{2}}&{{\bf{19}}}&{ - {\bf{4}}}&{{\bf{12}}}\\{ - {\bf{14}}}&{{\bf{11}}}&{ - {\bf{12}}}&{\bf{8}}\\{ - {\bf{2}}}&{{\bf{21}}}&{\bf{4}}&{\bf{8}}\end{array}} \right)\)

In Exercises 3-6, find (a) the maximum value of\(Q\left( {\rm{x}} \right)\)subject to the constraint\({{\rm{x}}^T}{\rm{x}} = 1\), (b) a unit vector\({\rm{u}}\)where this maximum is attained, and (c) the maximum of\(Q\left( {\rm{x}} \right)\)subject to the constraints\({{\rm{x}}^T}{\rm{x}} = 1{\rm{ and }}{{\rm{x}}^T}{\rm{u}} = 0\).

3.\(Q\left( x \right) = 5x_1^2 + 6x_2^2 + 7x_3^2 + 4x_1^{}x_2^{} - 4x_2^{}x_3^{}\).

Question: 5. Show that if v is an eigenvector of an \(n \times n\) matrix A and v corresponds to a nonzero eigenvalue of A, then v is in Col A. (Hint: Use the definition of an eigenvector.)

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.)
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