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Chapter 7: Q7.4-15E (page 395)

Question: Suppose the factorization below is an SVD of a matrix A, with the entries in U and Vrounded to two decimal places.

\(A = \left( {\begin{array}{*{20}{c}}{.{\bf{40}}}&{ - .{\bf{78}}}&{.{\bf{47}}}\\{.{\bf{37}}}&{ - .{\bf{33}}}&{ - .{\bf{87}}}\\{ - .{\bf{84}}}&{ - .{\bf{52}}}&{ - .{\bf{16}}}\end{array}} \right)\left( {\begin{array}{*{20}{c}}{{\bf{7}}{\bf{.10}}}&{\bf{0}}&{\bf{0}}\\{\bf{0}}&{{\bf{3}}{\bf{.10}}}&{\bf{0}}\\{\bf{0}}&{\bf{0}}&{\bf{0}}\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}{.{\bf{30}}}&{ - .{\bf{51}}}&{ - .{\bf{81}}}\\{.{\bf{76}}}&{.{\bf{64}}}&{ - .{\bf{12}}}\\{.{\bf{58}}}&{ - .{\bf{58}}}&{.{\bf{58}}}\end{array}} \right)\)

a. What is the rank of A?

b. Use this decomposition of A, with no calculations, to write a basis for ColA and a basis for NulA. (Hint: First write the columns of V.)

Short Answer

Expert verified

a. The rank of matrix A is 2.

b. The basis of ColA is \(\left\{ {\left( {\begin{array}{*{20}{c}}{.40}\\{.37}\\{ - .84}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .78}\\{ - .33}\\{ - .52}\end{array}} \right)} \right\}\) and the basis of Null space is \(\left( {\begin{array}{*{20}{c}}{.58}\\{ - .58}\\{.58}\end{array}} \right)\).

Step by step solution

01

Compare the given SVD to find the matrices

(a). On comparing the given SVD with the equation \(A = U\Sigma {V^T}\).

\(U = \left( {\begin{array}{*{20}{c}}{.40}&{ - .78}&{.47}\\{.37}&{ - .33}&{ - .87}\\{ - .84}&{ - .52}&{ - .16}\end{array}} \right)\), \(\Sigma = \left( {\begin{array}{*{20}{c}}{7.10}&0&0\\0&{3.10}&0\\0&0&0\end{array}} \right)\) and \(V = \left( {\begin{array}{*{20}{c}}{.30}&{.76}&{.58}\\{ - .51}&{.64}&{ - .58}\\{ - .81}&{ - .58}&{.58}\end{array}} \right)\)

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

02

Find the basis for NulA

(b). The orthogonal basis of column space of A is:

\(\left( {{{\bf{u}}_1},{{\bf{u}}_2}} \right) = \left\{ {\left( {\begin{array}{*{20}{c}}{.40}\\{.37}\\{ - .84}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .78}\\{ - .33}\\{ - .52}\end{array}} \right)} \right\}\)

The basis for Nullspace is:

\(\begin{array}{c}V = \left( {{{\bf{v}}_3}} \right)\\ = \left( {\begin{array}{*{20}{c}}{.58}\\{ - .58}\\{.58}\end{array}} \right)\end{array}\)

Thus, the basis of \({\rm{Col}}A\) is \(\left\{ {\left( {\begin{array}{*{20}{c}}{.40}\\{.37}\\{ - .84}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - .78}\\{ - .33}\\{ - .52}\end{array}} \right)} \right\}\) and the basis of Null space is \(\left( {\begin{array}{*{20}{c}}{.58}\\{ - .58}\\{.58}\end{array}} \right)\).

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Most popular questions from this chapter

Question 11: Prove that any \(n \times n\) matrix A admits a polar decomposition of the form \(A = PQ\), where P is a \(n \times n\) positive semidefinite matrix with the same rank as A and where Q is an \(n \times n\) orthogonal matrix. (Hint: Use a singular value decomposition, \(A = U\sum {V^T}\), and observe that \(A = \left( {U\sum {U^T}} \right)\left( {U{V^T}} \right)\).) This decomposition is used, for instance, in mechanical engineering to model the deformation of a material. The matrix P describe the stretching or compression of the material (in the directions of the eigenvectors of P), and Q describes the rotation of the material in space.

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.

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

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 \({\rm{Nul}}A\).

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.

15. \(\left( {\begin{aligned}{{}}{\,3}&4\\4&9\end{aligned}} \right)\)
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