Solved: In 19 and 20 verify the foregoing result for the

Chapter 8, Problem 19E

(choose chapter or problem)

Let P denote a matrix whose columns are eigenvectors \(\mathbf{K}_{1}, \quad \mathbf{K}_{2},\) . . . , \(\mathbf{K}_{n}\) corresponding to distinct eigenvalues \(\lambda_{1}, \lambda_{2}\), . . . , \(\lambda_{n}\) of an n x n matrix A. Then it can be shown that \(\mathbf{A}=\mathbf{P D P}^{-1}\), where D is a diagonal matrix defined by

\(\mathbf{D}=\left(\begin{array}{cccc}

\lambda_{1} & 0 & \cdots & 0 \\

0 & \lambda_{2} & \cdots & 0 \\

\cdot & \cdot & & \cdot \\

\cdot & \cdot & & \vdots \\

0 & 0 & \cdots & \lambda_{n}

\end{array}\right)

\)   (9)

In Problems 19 and 20 verify the foregoing result for the given matrix.

\(\mathbf{A}=\left(\begin{array}{rr}

2 & 1 \\

-3 & 6

\end{array}\right)

\)

Text Transcription:

mathbf K_1

mathbf K_2

mathbf K_n

lambda_1

lambda_2

lambda_n

mathbf A =mathbf P D P^-1

mathbf D = ({array} cccc lambda_1 & 0 & cdots & 0 0 & lambda_2 & cdots & 0 cdot & cdot & & cdot cdot & cdot & & vdots 0 & 0 & cdots & lambda_n {array})

mathbf A = ({array} rr 2 & 1 -3 & 6 {array})