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Chapter 7: Q7.4-16E (page 395)

Question: Repeat Exercise 15 for the following SVD of a \({\bf{3 \times 4}}\) matrix \(A\):

\(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{ - {\bf{.86}}}&{ - {\bf{.11}}}&{ - {\bf{.50}}}\\{{\bf{.31}}}&{{\bf{.68}}}&{ - {\bf{.67}}}\\{{\bf{.41}}}&{ - {\bf{.73}}}&{ - {\bf{5}}{\bf{.5}}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{\bf{12}}{\bf{.48}}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{{\bf{6}}{\bf{.34}}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\end{array}} \right){\bf{ \times }}\left( {\begin{array}{*{20}{c}}{{\bf{.66}}}&{ - {\bf{.03}}}&{ - {\bf{.35}}}&{{\bf{.66}}}\\{ - {\bf{1}}{\bf{.3}}}&{ - {\bf{.90}}}&{ - {\bf{.39}}}&{ - {\bf{.13}}}\\{{\bf{.65}}}&{{\bf{.08}}}&{ - {\bf{.16}}}&{ - {\bf{.73}}}\\{ - {\bf{.34}}}&{{\bf{.42}}}&{ - {\bf{8}}{\bf{.4}}}&{ - {\bf{0}}{\bf{.8}}}\end{array}} \right)\)

Short Answer

Expert verified
  1. The rank of the matrix is \(2\).
  2. \(COL\;A = \left\{ {\left( {\begin{array}{*{20}{c}}{ - .86}\\{.31}\\{.41}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .11}\\{.68}\\{ - .73}\end{array}} \right)} \right\}\) and \(Nul\;A = \left\{ {\left( {\begin{array}{*{20}{c}}{.65}\\{.08}\\{ - .16}\\{ - .73}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .34}\\{.42}\\{ - .84}\\{ - .08}\end{array}} \right)} \right\}\).

Step by step solution

01

(a) Find the rank of \(A\)

The given singular value decomposition is:

\(A = \left( {\begin{array}{*{20}{c}}{ - 0.86}&{ - 0.11}&{ - 0.50}\\{0.31}&{0.68}&{ - 0.67}\\{0.41}&{ - 0.73}&{ - 5.5}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{12.48}&0&0&0\\0&{6.34}&0&0\\0&0&0&0\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}{.66}&{ - .03}&{ - .35}&{.66}\\{ - 1.3}&{ - .90}&{ - .39}&{ - .13}\\{.65}&{.08}&{ - .16}&{ - .73}\\{ - .34}&{.42}&{ - 8.4}&{ - .08}\end{array}} \right)\)

Where the standard form of an SVD is\(A = U\sum {V^T}\).

Write the\(\sum \)matrix, which is the matrix of singular values of\(A\).

\(\sum = \left( {\begin{array}{*{20}{c}}{12.48}&0&0&0\\0&{6.34}&0&0\\0&0&0&0\end{array}} \right)\)

As the above matrix has two nonzero singular values, the rank of the matrix is \(2\).

02

(b) Write a basis for Col \(A\) and a basis for Nul \(A\)

From the SVD decomposition of the matrix\(A\) of rank\(r\), if,

\(U = \left( {{{\bf{u}}_1},{{\bf{u}}_2}, \ldots ,{{\bf{u}}_n}} \right)\)is the left singular vector.

\(V = \left( {{{\bf{v}}_1},{{\bf{v}}_2}, \ldots ,{{\bf{v}}_n}} \right)\)is the right singular vector and

\({\rm{\Sigma }} = \left( {{\sigma _1},{\sigma _2}, \ldots \ldots ,{\sigma _n}} \right)\)the singular values.

Since\(({{\bf{u}}_1},{{\bf{u}}_2}, \ldots \ldots ,{{\bf{u}}_r})\)is an orthogonal basis for col\(A\)and\(\left\{ {{{\bf{v}}_{r + 1}}, \ldots \ldots ,{{\bf{v}}_n}} \right\}\)is an orthogonal basis for Nul\(A\).

The matrix\(V\)is

\(V = \left( {\begin{array}{*{20}{c}}{.66}&{ - .13}&{.65}&{ - .34}\\{ - .03}&{ - .90}&{.08}&{.42}\\{ - .35}&{ - .39}&{ - .16}&{ - .84}\\{.66}&{ - .13}&{ - .73}&{ - .08}\end{array}} \right)\)

Thus, the basis for Column space is:

\(\begin{array}{c}COL\;A = \left\{ {{{\bf{u}}_1},{{\bf{u}}_2}} \right\}\\ = \left\{ {\left( {\begin{array}{*{20}{c}}{ - .86}\\{.31}\\{.41}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .11}\\{.68}\\{ - .73}\end{array}} \right)} \right\}\end{array}\)

Thus, the basis for Null space is:

\(\begin{array}{c}Nul\;A = \left\{ {{{\bf{v}}_1},{{\bf{v}}_2}} \right\}\\ = \left\{ {\left( {\begin{array}{*{20}{c}}{.65}\\{.08}\\{ - .16}\\{ - .73}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .34}\\{.42}\\{ - .84}\\{ - .08}\end{array}} \right)} \right\}\end{array}\)

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Most popular questions from this chapter

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.

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

Determine which of the matrices in Exercises 7鈥�12 are orthogonal. If orthogonal, find the inverse.

9. \(\left[ {\begin{aligned}{{}}{ - 4/5}&{\,\,\,3/5}\\{3/5}&{\,\,4/5}\end{aligned}} \right]\)

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.)

Question: Mark Each statement True or False. Justify each answer. In each part, A represents an \(n \times n\) matrix.

  1. If A is orthogonally diagonizable, then A is symmetric.
  2. If A is an orthogonal matrix, then A is symmetric.
  3. If A is an orthogonal matrix, then \(\left\| {A{\bf{x}}} \right\| = \left\| {\bf{x}} \right\|\) for all x in \({\mathbb{R}^n}\).
  4. The principal axes of a quadratic from \({{\bf{x}}^T}A{\bf{x}}\) can be the columns of any matrix P that diagonalizes A.
  5. If P is an \(n \times n\) matrix with orthogonal columns, then \({P^T} = {P^{ - {\bf{1}}}}\).
  6. If every coefficient in a quadratic form is positive, then the quadratic form is positive definite.
  7. If \({{\bf{x}}^T}A{\bf{x}} > {\bf{0}}\) for some x, then the quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is positive definite.
  8. By a suitable change of variable, any quadratic form can be changed into one with no cross-product term.
  9. The largest value of a quadratic form \({{\bf{x}}^T}A{\bf{x}}\), for \(\left\| {\bf{x}} \right\| = {\bf{1}}\) is the largest entery on the diagonal A.
  10. The maximum value of a positive definite quadratic form \({{\bf{x}}^T}A{\bf{x}}\) is the greatest eigenvalue of A.
  11. A positive definite quadratic form can be changed into a negative definite form by a suitable change of variable \({\bf{x}} = P{\bf{u}}\), for some orthogonal matrix P.
  12. An indefinite quadratic form is one whose eigenvalues are not definite.
  13. If P is an \(n \times n\) orthogonal matrix, then the change of variable \({\bf{x}} = P{\bf{u}}\) transforms \({{\bf{x}}^T}A{\bf{x}}\) into a quadratic form whose matrix is \({P^{ - {\bf{1}}}}AP\).
  14. If U is \(m \times n\) with orthogonal columns, then \(U{U^T}{\bf{x}}\) is the orthogonal projection of x onto ColU.
  15. If B is \(m \times n\) and x is a unit vector in \({\mathbb{R}^n}\), then \(\left\| {B{\bf{x}}} \right\| \le {\sigma _{\bf{1}}}\), where \({\sigma _{\bf{1}}}\) is the first singular value of B.
  16. A singular value decomposition of an \(m \times n\) matrix B can be written as \(B = P\Sigma Q\), where P is an \(m \times n\) orthogonal matrix and \(\Sigma \) is an \(m \times n\) diagonal matrix.
  17. If A is \(n \times n\), then A and \({A^T}A\) have the same singular values.

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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