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Q13SE

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Linear Algebra and its Applications
Found in: Page 395
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Question: 13. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Suppose the equation \(A{\rm{x}} = {\rm{b}}\) is consistent, and let \({{\rm{x}}^ + } = {A^ + }{\rm{b}}\). By Exercise 23 in Section 6.3, there is exactly one vector \({\rm{p}}\) in Row \(A\) such that \(A{\rm{p}} = {\rm{b}}\). The following steps prove that \({{\rm{x}}^ + } = {\rm{p}}\) and \({{\rm{x}}^ + }\)is the minimum length solution of \(A{\rm{x}} = {\rm{b}}\).

  1. Show that \({{\rm{x}}^ + }\) is in Row \(A\). (Hint: Write \({\rm{b}}\) as \(A{\rm{x}}\) for some \({\rm{x}}\), and use Exercise 12.)
  2. Show that \({{\rm{x}}^ + }\) is a solution of \(A{\rm{x}} = {\rm{b}}\).
  3. Show that if \({\rm{u}}\) is any solution of \(A{\rm{x}} = {\rm{b}}\), then \(\left\| {{{\rm{x}}^ + }} \right\| \le \left\| {\rm{u}} \right\|\), with equality only if \({\rm{u}} = {{\rm{x}}^ + }\).

  1. It is shown that \({{\rm{x}}^ + }\) is in Row \(A\).
  2. It is shown that \({{\rm{x}}^ + }\) is the solution of \(A{\rm{x}} = {\rm{b}}\).
  3. It is shown that \(\left\| {{{\rm{x}}^ + }} \right\| < \left\| {\rm{u}} \right\|\), and equality holds if \({\bf{u}} = {{\rm{x}}^ + }\).
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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Simplify for vector \({{\rm{x}}^ + }\)

It is given that the system \(A{\rm{x}} = {\rm{b}}\) is consistent and \({{\rm{x}}^ + } = {A^ + }{\rm{b}}\), then the system \(A{\rm{x}} = {\rm{b}}\) can be simplified as follows:

\(\begin{array}{c}{{\rm{x}}^ + } = {A^ + }{\rm{b}}\\ = {A^ + }A{\rm{x}}\end{array}\)

As \({{\rm{x}}^ + } = {A^ + }A{\rm{x}}\), so\({{\rm{x}}^ + }\) is the orthogonal projection of \({\rm{x}}\) onto Row \(A\).

Step 2: Simplify for product \(A{{\rm{x}}^ + }\)

Substitute \({{\rm{x}}^ + } = {A^ + }A{\rm{x}}\), into \(A{{\rm{x}}^ + }\), and simplify using \(A{\rm{x}} = {\rm{b}}\), as follows:

\(\begin{array}{c}A{{\rm{x}}^ + } = A\left( {{A^ + }A{\rm{x}}} \right)\\ = A{A^ + }A{\rm{x}}\\ = A{\rm{x}}\\ = {\rm{b}}\end{array}\)

Thus, it is shown that \({{\rm{x}}^ + }\) is the solution of \(A{\rm{x}} = {\rm{b}}\).

Step 3: Prove \(\left\| {{{\rm{x}}^ + }} \right\| < \left\| {\rm{u}} \right\|\)

Suppose the system \(A{\rm{x}} = {\rm{b}}\) is satisfied by another basis \({\rm{u}}\), such that, \(A{\rm{u}} = {\rm{b}}\). It is also known that \({{\rm{x}}^ + }\)is the orthogonal projection of \({\rm{x}}\) onto Row \(A\).

Apply the Pythagorean theorem on \({\left\| {\rm{u}} \right\|^2}\), as follows:

\(\begin{array}{c}{\left\| {\rm{u}} \right\|^2} = {\left\| {{{\rm{x}}^ + }} \right\|^2} + {\left\| {{\rm{u}} - {{\rm{x}}^ + }} \right\|^2}\\ \ge {\left\| {{{\rm{x}}^ + }} \right\|^2}\end{array}\)

So \(\left\| {{{\rm{x}}^ + }} \right\| < \left\| {\rm{u}} \right\|\), and equality holds if \({\bf{u}} = {{\rm{x}}^ + }\).

Most popular questions for Math Textbooks

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.

Question: 14. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Given any \({\rm{b}}\) in \({\mathbb{R}^m}\), adapt Exercise 13 to show that \({A^ + }{\rm{b}}\) is the least-squares solution of minimum length. [Hint: Consider the equation \(A{\rm{x}} = {\rm{b}}\), where \(\mathop {\rm{b}}\limits^\^ \) is the orthogonal projection of \({\rm{b}}\) onto Col \(A\).

In Exercises 1 and 2, find the change of variable \({\rm{x}} = P{\rm{y}}\) that transforms the quadratic form \({{\rm{x}}^T}A{\rm{x}}\) into \({{\rm{y}}^T}D{\rm{y}}\) as shown.

2. \(3x_1^2 + 3x_2^2 + 5x_3^2 + 6x_1^{}x_2^{} + 2x_1^{}x_3^{} + 2x_2^{}x_3^{} = 7y_1^2 + 4y_2^2\).

Question: 12. Exercises 12–14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Verify the properties of \({A^ + }\):

a. For each \({\rm{y}}\) in \({\mathbb{R}^m}\), \(A{A^ + }{\rm{y}}\) is the orthogonal projection of \({\rm{y}}\) onto \({\rm{Col}}\,A\).

b. For each \({\rm{x}}\) in \({\mathbb{R}^n}\), \({A^ + }A{\rm{x}}\) is the orthogonal projection of \({\rm{x}}\) onto \({\rm{Row}}\,A\).

c. \(A{A^ + }A = A\)and \({A^ + }A{A^ + } = {A^ + }\).

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

10. \(\left( {\begin{aligned}{{}}{1/3}&{\,\,2/3}&{\,\,2/3}\\{2/3}&{\,\,1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\)
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