/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q4E Determine which of the matrices ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q4E (page 395)

Determine which of the matrices in Exercises 1鈥�6 are symmetric.

4. \(\left( {\begin{aligned}{{}}0&8&3\\8&0&{ - 4}\\3&2&0\end{aligned}} \right)\)

Short Answer

Expert verified

The given matrix is not symmetric.

Step by step solution

01

Find the transpose

A matrix\(A\) with, \(n \times n\) dimension, is symmetric if it satisfies the equation\({A^T} = A\).

It is given that\(A = \left( {\begin{aligned}{{}}0&8&3\\8&0&{ - 4}\\3&2&0\end{aligned}} \right)\). Find the transpose of\(A\), as shown below:

\(\begin{aligned}{}{A^T} = \left( {\begin{aligned}{{}}0&8&3\\8&0&2\\3&{ - 4}&0\end{aligned}} \right)\\ \ne A\end{aligned}\)

02

Draw the conclusion

As \({A^T} \ne A\), so it can be concluded that \(A\) is not asymmetric matrix.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let u be a unit vector in \({\mathbb{R}^n}\), and let \(B = {\bf{u}}{{\bf{u}}^T}\).

  1. Given any x in \({\mathbb{R}^n}\), compute Bx and show that Bx is the orthogonal projection of x onto u, as described in Section 6.2.
  2. Show that B is a symmetric matrix and \({B^{\bf{2}}} = B\).
  3. Show that u is an eigenvector of B. What is the corresponding eigenvalue?

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

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

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.

13. \({\bf{ - }}x_{\bf{1}}^{\bf{2}}{\bf{ - 6}}{x_{\bf{1}}}{x_{\bf{2}}} + {\bf{9}}x_{\bf{2}}^{\bf{2}}\)

Question: 12. Exercises 12鈥�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 \({A^ + }A{A^ + } = {A^ + }\).

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).

9. Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A. (See the Exercises in Section 6.5.)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.