find the Basis of eigen space A = 3ైణ అడా

Consider the given matrix. A=0-333-536-64 Find the Eigen values of the matrix.…

Answered: A = 3 ైణ అ డా

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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find the Basis of eigen space

Expert Solution

Step 1

Consider the given matrix.

$A = [\begin{matrix} 0 & - 3 & 3 \\ 3 & - 5 & 3 \\ 6 & - 6 & 4 \end{matrix}]$

Find the Eigen values of the matrix.

$\begin{array}{rcl} A - λ I & = & 0 \\ [\begin{matrix} 0 & - 3 & 3 \\ 3 & - 5 & 3 \\ 6 & - 6 & 4 \end{matrix}] - [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] λ & = & 0 \\ [\begin{matrix} - λ & - 3 & 3 \\ 3 & - 5 - λ & 3 \\ 6 & - 6 & 4 - λ \end{matrix}] & = & 0 \end{array}$

Step 2

Further simplify.

$\begin{array}{rcl} - λ [(- 5 - λ) (4 - λ) + 18] + 3 [3 (4 - λ) - 18] + 3 [- 18 + 6 (5 + λ)] & = & 0 \\ [- λ (- 5 - λ) (4 - λ) - λ 18] + 9 λ + 18 & = & 0 \\ - λ^{3} - λ^{2} + 11 λ + 18 & = & 0 \\ (λ + 2) (- λ^{2} + λ + 9) & = & 0 \\ λ & = & - 2 \\ λ & = & \frac{1 \pm \sqrt{37}}{2} \end{array}$

The Eigen values are calculated above.

Step 3

Now, find the basis for the Eigen values.

For $λ = - 2$ .

$\begin{array}{rcl} A - I λ_{1} & = & 0 \\ [\begin{matrix} 2 & - 3 & 3 \\ 3 & - 3 & 3 \\ 6 & - 6 & 6 \end{matrix}] & = & 0 \end{array}$

Now, find the echelon form.

$\begin{array}{rcl} [\begin{matrix} 2 & - 3 & 3 \\ 3 & - 3 & 3 \\ 6 & - 6 & 6 \end{matrix}] \end{array} \overset{R_{3} = R_{3} - 3 R_{1}}{\to} \begin{array}{rcl} [\begin{matrix} 2 & - 3 & 3 \\ 3 & - 3 & 3 \\ 0 & 3 & - 3 \end{matrix}] \end{array} \overset{R_{2} = R_{2} - \frac{3}{2} R_{1}}{\to} \begin{array}{rcl} [\begin{matrix} 2 & - 3 & 3 \\ 3 & \frac{3}{2} & - \frac{3}{2} \\ 0 & 3 & - 3 \end{matrix}] \end{array} \overset{R_{3} = R_{3} - 2 R_{2}}{\to} \begin{array}{rcl} [\begin{matrix} 2 & - 3 & 3 \\ 3 & \frac{3}{2} & - \frac{3}{2} \\ 0 & 0 & 0 \end{matrix}] \end{array}$

Step 4

Further simplify.

$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - 1 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

Here, $v_{3} = t$ and $v_{1} = 0, v_{2} = t$ .

$\begin{array}{l} V = [\begin{matrix} 0 \\ t \\ t \end{matrix}] \\ = [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}] t \end{array}$

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