Math / Differential Equations and Linear Algebra 2 / Chapter 5.3 / Problem 50

Differential Equations and Linear Algebra | 2nd Edition | ISBN: 9780131860612 | Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly West

?Orthogonal Eigenvectors Let A be a symmetric matrix (that is, A = AT) with distinct

Chapter 5, Problem 50

(choose chapter or problem)

Orthogonal Eigenvectors Let A be a symmetric matrix (that is, A = AT) with distinct eigenvalues λ1 and λ2. For such a matrix, if $\bar{v}_{1} \text { and } \bar{v}_{2}$ are eigenvectors belonging to the distinct eigenvalues λ1 and λ2, respectively, then $\bar{v}_{1} \text { and } \bar{v}_{2}$ are orthogonal

(a) Illustrate this for

$A=\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right]$

(b) Prove fact for an n times n symmetric matrix. Use the fact that $\bar{v}_{1}, \bar{v}_{2}=\overline{v_{1}^{T}} \bar{v}_{2}$ (as a matrix product).

Equation Transcription:

${\overline{v}}_{1}\ and\ {\overline{v}}_{2}$

$A=\left[{{\ }_{2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 4}^{1\ \ \ \ \ \ \ \ \ \ \ \ \ \ 2}}\right]$

${\overline{v}}_{1},\ {\overline{v}}_{2}=\overline{{v}_{1}^{T}}{\overline{v}}_{2}$

Text Transcription:

bar v_1 and bar v_2

A=[1 2 _ 2 4]

bar v_1 , bar v_2 = bar v_1 ^T bar v_2

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