Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 25E from Chapter 7.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 25E
Chapter:
Problem:
In Exercises 25 and 26, mark each statement True or False. Justify each answer. a. An n × n matrix that is orthogonally diagonalizable must be symmetric. c. An n × n symmetric matrix has n distinct real eigenvalues. d. For a nonzero v in Rn, the matrix vvT is called a projection matrix.
Step-by-Step Solution
Given Information
We are given with some statements that we have check whether they are True or False
Step-1: (a)
By Theorem-2: An $n \times n$ matrix A is orthogonally diagonalizable if and only if A is a symmetric matrix. Therefore,
The statement is True
Step-2: (b)
if $A^T = A$, then the matrix is symmetric. Also, u and v are eigenvectors of the matrix.
We know that any two eigenvectors from different Eigenspaces are orthogonal. Thus, \[ \mathbf { u } \cdot \mathbf { v } = 0 \]
The statement is True
Step-3: (c)
An $n \times n$ symmetric matrix has $n$ distinct real eigenvalues. However, it is not necessary that they have to be distinct. Therefore,
The statement is False
Step-4: (d)
By theorem, If $v$ is a unit matrix in $R^n$ then matrix $vv^T$ is called a projection matrix. The statement misses the term unit vector, Therefore
The statement is False
We are given with some statements that we have check whether they are True or False
Step-1: (a)
By Theorem-2: An $n \times n$ matrix A is orthogonally diagonalizable if and only if A is a symmetric matrix. Therefore,
The statement is True
Step-2: (b)
if $A^T = A$, then the matrix is symmetric. Also, u and v are eigenvectors of the matrix.
We know that any two eigenvectors from different Eigenspaces are orthogonal. Thus, \[ \mathbf { u } \cdot \mathbf { v } = 0 \]
The statement is True
Step-3: (c)
An $n \times n$ symmetric matrix has $n$ distinct real eigenvalues. However, it is not necessary that they have to be distinct. Therefore,
The statement is False
Step-4: (d)
By theorem, If $v$ is a unit matrix in $R^n$ then matrix $vv^T$ is called a projection matrix. The statement misses the term unit vector, Therefore
The statement is False