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Algebra and Geometry: Introduction to Linear Algebra for Science & Engineering

Daniel Norman And Dan Wolczk

Chapter 8

Symmetric Matrices and Quadratic Forms - all with Video Answers

Educators

Section 1

Diagonalization of Symmetric Matrices

02:18

Problem 1

Determine which of the following matrices are symmetric.
(a) $A=\left[\begin{array}{rr}0 & 2 \\ 2 & -1\end{array}\right]$
(b) $B=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$
(c) $C=\left[\begin{array}{rrr}1 & 2 & 1 \\ -2 & 1 & 2 \\ -1 & -2 & 1\end{array}\right]$
(d) $D=\left[\begin{array}{rrr}0 & -1 & 1 \\ -1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]$

Carole Wastog
Carole Wastog
Numerade Educator
14:15

Problem 2

For each of the following symmetric matrices, find an orthogonal matrix $P$ and diagonal matrix $D$ such that $P^{T} A P=D$.
(a) $A=\left[\begin{array}{rr}1 & -2 \\ -2 & 1\end{array}\right]$
(b) $A=\left[\begin{array}{rr}5 & 3 \\ 3 & -3\end{array}\right]$
(c) $A=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0\end{array}\right]$
(d) $A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & -1 & -2 \\ -2 & -2 & 0\end{array}\right]$
(e) $A=\left[\begin{array}{rrr}1 & 8 & 4 \\ 8 & 1 & -4 \\ 4 & -4 & 7\end{array}\right]$

Chris Trentman
Chris Trentman
Numerade Educator