Thread: Finding matrices with a given eigenvector/eigenvalue

1. Finding matrices with a given eigenvector/eigenvalue

Problem: Find all 2x2 matrices for which $\begin{bmatrix} 2\\ 3 \end{bmatrix}$ is an eigenvector with associated eigenvalue -1.

Here's my attempt:

$A \vec{v} = \lambda \vec{v}$

$A\begin{bmatrix} 2\\ 3 \end{bmatrix} = (-1)\begin{bmatrix} 2\\ 3 \end{bmatrix}$

$\begin{bmatrix} a & b\\ c & d \end{bmatrix}\begin{bmatrix} 2\\ 3 \end{bmatrix} = (-1)\begin{bmatrix} 2\\ 3 \end{bmatrix}$

$\begin{bmatrix} a & b\\ c & d \end{bmatrix}\begin{bmatrix} 2\\ 3 \end{bmatrix} = \begin{bmatrix} -2\\ -3 \end{bmatrix}$

$\begin{bmatrix} 2a + 3b\\ 2c + 3d \end{bmatrix} = \begin{bmatrix} -2\\ -3 \end{bmatrix}$

This is where I start to get off track, assuming I was actually on the correct track. A push in the right direction would be great.

Thanks.

2. Originally Posted by tangibleLime
Problem: Find all 2x2 matrices for which $\begin{bmatrix} 2\\ 3 \end{bmatrix}$ is an eigenvector with associated eigenvalue -1.

Here's my attempt:

$A \vec{v} = \lambda \vec{v}$

$A\begin{bmatrix} 2\\ 3 \end{bmatrix} = (-1)\begin{bmatrix} 2\\ 3 \end{bmatrix}$

$\begin{bmatrix} a & b\\ c & d \end{bmatrix}\begin{bmatrix} 2\\ 3 \end{bmatrix} = (-1)\begin{bmatrix} 2\\ 3 \end{bmatrix}$

$\begin{bmatrix} a & b\\ c & d \end{bmatrix}\begin{bmatrix} 2\\ 3 \end{bmatrix} = \begin{bmatrix} -2\\ -3 \end{bmatrix}$

$\begin{bmatrix} 2a + 3b\\ 2c + 3d \end{bmatrix} = \begin{bmatrix} -2\\ -3 \end{bmatrix}$

This is where I start to get off track, assuming I was actually on the correct track. A push in the right direction would be great.

Thanks.
One hundred percent correct, actually! Just solve those last two simultaneous equations!

3. 2a+3b=-2

b=-(2a+2)/3

2c+3d=-3

d=-(2c+3)/3

4. Huzzah! Thanks, I got it.

And thanks to alexmahone for confirming my answer.

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how to find the matrix if its eigen values and eigen vectors are given

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