Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 7

Symmetric Matrices and Quadratic Forms - all with Video Answers

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Section 1

Diagonalization of Symmetric Matrices

Problem 1

Determine which of the matrices in Exercises $1-6$ are symmetric.
$$
\left[\begin{array}{rr}{3} & {5} \\ {5} & {-7}\end{array}\right]
$$

Chris T.

Chris T.

Numerade Educator

Problem 2

Determine which of the matrices in Exercises $1-6$ are symmetric.
$$
\left[\begin{array}{rr}{3} & {-5} \\ {-5} & {-3}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 3

Determine which of the matrices in Exercises $1-6$ are symmetric.
$$
\left[\begin{array}{ll}{2} & {3} \\ {2} & {4}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 4

Determine which of the matrices in Exercises $1-6$ are symmetric.
$$
\left[\begin{array}{rrr}{0} & {8} & {3} \\ {8} & {0} & {-4} \\ {3} & {2} & {0}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 5

Determine which of the matrices in Exercises $1-6$ are symmetric.
$$
\left[\begin{array}{rrr}{-6} & {2} & {0} \\ {2} & {-6} & {2} \\ {0} & {2} & {-6}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 6

Determine which of the matrices in Exercises $1-6$ are symmetric.
$$
\left[\begin{array}{cccc}{1} & {2} & {2} & {1} \\ {2} & {2} & {2} & {1} \\ {2} & {2} & {1} & {2}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 7

Determine which of the matrices in Exercises $7-12$ are orthogonal. If orthogonal, find the inverse.
$$
\left[\begin{array}{rr}{.6} & {.8} \\ {.8} & {-.6}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 8

Determine which of the matrices in Exercises $7-12$ are orthogonal. If orthogonal, find the inverse.
$$
\left[\begin{array}{rr}{1} & {1} \\ {1} & {-1}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 9

Determine which of the matrices in Exercises $7-12$ are orthogonal. If orthogonal, find the inverse.
$$
\left[\begin{array}{rr}{-4 / 5} & {3 / 5} \\ {3 / 5} & {4 / 5}\end{array}\right]
$$

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Problem 10

Determine which of the matrices in Exercises $7-12$ are orthogonal. If orthogonal, find the inverse.
$$
\left[\begin{array}{ccc}{1 / 3} & {2 / 3} & {2 / 3} \\ {2 / 3} & {1 / 3} & {-2 / 3} \\ {2 / 3} & {-2 / 3} & {1 / 3}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 11

Determine which of the matrices in Exercises $7-12$ are orthogonal. If orthogonal, find the inverse.
$$
\left[\begin{array}{ccc}{2 / 3} & {2 / 3} & {1 / 3} \\ {0} & {1 / 3} & {-2 / 3} \\ {5 / 3} & {-4 / 3} & {-2 / 3}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 12

Determine which of the matrices in Exercises $7-12$ are orthogonal. If orthogonal, find the inverse.
$$
\left[\begin{array}{rrrr}{.5} & {.5} & {-.5} & {-.5} \\ {.5} & {.5} & {.5} & {.5} \\ {.5} & {-.5} & {-.5} & {.5} \\ {.5} & {-.5} & {.5} & {-.5}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 13

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{ll}{3} & {1} \\ {1} & {3}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 14

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{rr}{1} & {-5} \\ {-5} & {1}\end{array}\right]
$$

Bobby B.

University of North Texas

Problem 15

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{ll}{3} & {4} \\ {4} & {9}\end{array}\right]
$$

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Problem 16

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{rr}{6} & {-2} \\ {-2} & {9}\end{array}\right]
$$

Bobby B.

University of North Texas

Problem 17

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{lll}{1} & {1} & {5} \\ {1} & {5} & {1} \\ {5} & {1} & {1}\end{array}\right]
$$

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Problem 18

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{rrr}{3} & {-2} & {4} \\ {-2} & {6} & {2} \\ {4} & {2} & {3}\end{array}\right]
$$

Bobby B.

University of North Texas

Problem 19

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{rrr}{3} & {-2} & {4} \\ {-2} & {6} & {2} \\ {4} & {2} & {3}\end{array}\right]
$$

Anthony R.

Numerade Educator

Problem 20

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{rrr}{5} & {8} & {-4} \\ {8} & {5} & {-4} \\ {-4} & {-4} & {-1}\end{array}\right]
$$

Pagadala K.

Numerade Educator

Problem 21

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{llll}{4} & {3} & {1} & {1} \\ {3} & {4} & {1} & {1} \\ {1} & {1} & {4} & {3} \\ {1} & {1} & {3} & {4}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 22

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: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{cccc}{4} & {0} & {1} & {0} \\ {0} & {4} & {0} & {1} \\ {1} & {0} & {4} & {0} \\ {0} & {1} & {0} & {4}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 23

Let $A=\left[\begin{array}{rrr}{4} & {-1} & {-1} \\ {-1} & {4} & {-1} \\ {-1} & {-1} & {4}\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}{1} \\ {1} \\ {1}\end{array}\right] .$ Verify that 5 is an eigenvalue of $A$ and $\mathbf{v}$ is an eigenvector. Then orthogonally diagonalize $A .$

Chris T.

Numerade Educator

Problem 24

Let $A=\left[\begin{array}{rrr}{2} & {-1} & {1} \\ {-1} & {2} & {-1} \\ {1} & {-1} & {2}\end{array}\right], \quad \mathbf{v}_{1}=\left[\begin{array}{r}{-1} \\ {0} \\ {1}\end{array}\right], \quad$ and $\quad \mathbf{v}_{2}=$ $\left[\begin{array}{r}{1} \\ {-1} \\ {1}\end{array}\right] .$ Verify that $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are eigenvectors of $A .$ Then orthogonally diagonalize $A$

Chris T.

Numerade Educator

Problem 25

In Exercises 25 and $26,$ mark each statement True or False. Justify each answer.
a. An $n \times n$ matrix that is orthogonally diagonalizable must be symmetric.
b. If $A^{T}=A$ and if vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $A \mathbf{u}=3 \mathbf{u}$ and $A \mathbf{v}=4 \mathbf{v},$ then $\mathbf{u} \cdot \mathbf{v}=0$
c. $\operatorname{An} n \times n$ symmetric matrix has $n$ distinct real eigenval-
ues.
d. For a nonzero $\mathbf{v}$ in $\mathbb{R}^{n},$ the matrix $\mathbf{v}^{T}$ is called a projection matrix.

Chris T.

Numerade Educator

Problem 26

In Exercises 25 and $26,$ mark each statement True or False. Justify each answer.
a. There are symmetric matrices that are not orthogonally diagonalizable.
b. If $B=P D P^{T}$ , where $P^{T}=P^{-1}$ and $D$ is a diagonal
matrix, then $B$ is a symmetric matrix.
c. An orthogonal matrix is orthogonally diagonalizable. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Chris T.

Numerade Educator

Problem 27

Show that if $A$ is an $n \times n$ symmetric matrix, then $(A \mathbf{x}) \cdot \mathbf{y}=$
$\mathbf{x} \cdot(A \mathbf{y})$ for all $\mathbf{x}, \mathbf{y}$ in $\mathbb{R}^{n}$ .

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Problem 28

Suppose $A$ is a symmetric $n \times n$ matrix and $B$ is any $n \times m$ matrix. Show that $B^{T} A B, B^{T} B,$ and $B B^{T}$ are symmetric matrices.

Pagadala K.

Numerade Educator

Problem 29

Suppose $A$ is invertible and orthogonally diagonalizable. Explain why $A^{-1}$ is also orthogonally diagonalizable.

Anya J.

Numerade Educator

Problem 30

Suppose $A$ and $B$ are both orthogonally diagonalizable and $A B=B A .$ Explain why $A B$ is also orthogonally diagonalizable.

Chris T.

Numerade Educator

Problem 31

Let $A=P D P^{-1},$ where $P$ is orthogonal and $D$ is diagonal, and let $\lambda$ be an eigenvalue of $A$ of multiplicity $k$ . Then $\lambda$ appears $k$ times on the diagonal of $D .$ Explain why the
dimension of the eigenspace for $\lambda$ is $k$

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Problem 32

Suppose $A=P R P^{-1},$ where $P$ is orthogonal and $R$ is upper triangular. Show that if $A$ is symmetric, then $R$ is symmetric and hence is actually a diagonal matrix.

Chris T.

Numerade Educator

Problem 33

Construct a spectral decomposition of $A$ from Example 2

P K.

Numerade Educator

Problem 34

Construct a spectral decomposition of $A$ from Example 3

Chris T.

Numerade Educator

Problem 35

Let $\mathbf{u}$ be a unit vector in $\mathbb{R}^{n},$ and let $B=\mathbf{u} \mathbf{u}^{T}$
a. Given any $\mathbf{x}$ in $\mathbb{R}^{n}$ , compute $B \mathbf{x}$ and show that $B \mathbf{x}$ is the orthogonal projection of $\mathbf{x}$ onto $\mathbf{u},$ as described in
Section $6.2 .$
b. Show that $B$ is a symmetric matrix and $B^{2}=B$ .
c. Show that $B$ is an eigenvector of $B .$ What is the corresponding eigenvalue?

Chris T.

Numerade Educator

Problem 36

Let $B$ be an $n \times n$ symmetric matrix such that $B^{2}=B .$ Any such matrix is called a projection matrix (or an orthogonal projection matrix). Given any $y$ in $\mathbb{R}^{n},$ let $\hat{\mathbf{y}}=B \mathbf{y}$ and $\mathbf{z}=\mathbf{y}-\hat{\mathbf{y}}$
a. Show that $\mathbf{z}$ is orthogonal to $\hat{\mathbf{y}}$
b. Let $W$ be the column space of $B$ . Show that $y$ is the sum of a vector in $W$ and a vector in $W^{\perp} .$ Why does this prove that $B y$ is the orthogonal projection of $y$ onto the column
space of $B ?$

Chris T.

Numerade Educator

Problem 37

[M] Orthogonally 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 orthonormal basis for $\mathrm{Nul}(A-\lambda I),$ as in Examples 2 and $3 .$
$$
\left[\begin{array}{rrrr}{6} & {2} & {9} & {-6} \\ {2} & {6} & {-6} & {9} \\ {9} & {-6} & {6} & {2} \\ {-6} & {9} & {2} & {6}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 38

[M] Orthogonally 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 orthonormal basis for $\mathrm{Nul}(A-\lambda I),$ as in Examples 2 and $3 .$
$$
\left[\begin{array}{rrrr}{.63} & {-.18} & {-.06} & {-.04} \\ {-.18} & {.84} & {-.04} & {.12} \\ {-.06} & {-.04} & {.72} & {-.12} \\ {-.04} & {.12} & {-.12} & {.66}\end{array}\right]
$$

Chris T.

Numerade Educator

Problem 39

[M] Orthogonally 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 orthonormal basis for $\mathrm{Nul}(A-\lambda I),$ as in Examples 2 and $3 .$
$$
\left[\begin{array}{rrrr}{31} & {.58} & {.08} & {.44} \\ {.58} & {-.56} & {.44} & {-.58} \\ {.08} & {.44} & {.19} & {-.08} \\ {.44} & {-.58} & {-.08} & {.31}\end{array}\right]
$$

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Problem 40

[M] Orthogonally 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 orthonormal basis for $\mathrm{Nul}(A-\lambda I),$ as in Examples 2 and $3 .$
$$
\left[\begin{array}{rrrrr}{8} & {2} & {2} & {-6} & {9} \\ {2} & {8} & {2} & {-6} & {9} \\ {2} & {2} & {8} & {-6} & {9} \\ {-6} & {-6} & {-6} & {24} & {9} \\ {9} & {9} & {9} & {9} & {-21}\end{array}\right]
$$

Chris T.

Numerade Educator