Eigenvectors of $\left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$

Question

I calculated the eigenvalues of the following matrix to be $a$ and $-b$.

$J = \left( \begin{array}{ccc} a & 0 \\ 0 & -b \end{array} \right)$

But when I use the formula $(J - \lambda I)v = 0$ with either $a$ or $-b$ as eigenvalue to calculate the eigenvectors I get an indeterminate system of equations as they cancel with the $a$ and $-b$ in the matrix.

Yet Wolfram Alpha says the eigenvectors are $(1, 0)$ and $(0, 1)$?

Answer 1

if $a=b=0$

Then you have

$$\begin{pmatrix}a&0\\0&-b\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix}.$$

This means that every basis $(v_1,v_2)$ of $\Bbb{R}^2$ is a basis of eigenvectors. One can verify this by seeing that the dimension of the kernel of $A$ is $2$ and the fact that multiplying the zero matrix by $v_1$ and $v_2$ we get the vector $(0,0)^T$.

If $a=-b$ Then we have

$$(J-aI)=\begin{pmatrix}0&0\\0&-b-a\end{pmatrix}=\begin{pmatrix}0&0\\0&-b+b\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix}.$$

You can do the same reasoning as above to conclude that every basis of $\Bbb{R}^2$ is a basis of eigenvectors.

If $a\neq -b$ we have:

$$(J-aI)v_1=0 \iff\begin{pmatrix}0&0\\0&-b-a\end{pmatrix}v_1=0\implies v_1 = k\cdot \begin{pmatrix}1\\0\end{pmatrix},\quad k\in \Bbb{R},$$

because $$\begin{pmatrix}0&0\\0&-b-a\end{pmatrix} \begin{pmatrix}k\\0\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}.$$

Do you get the second eigenvector?

Answer 2

One way to answer this question is to remember the definition of an eigenvector.

An eigenvector $v$ of a matrix $A$ is a vector which satisfied $Av=\lambda v$ for some scalar $\lambda$.

So we can guess and check. My guesses would be $\left(\begin{array}{c} 1\\0\end{array}\right)$ and $\left(\begin{array}{c} 0\\1\end{array}\right)$. And a simple check would verify that these are indeed eigenvectors.

Guess and check is a very reasonable way to solve a math problem.

Answer 3

There's two cases:

• if $a=-b$ then the given matrix is $aI_2$ and then any non zero vector $v$ is an eigenvector for $J$
• if $a\ne-b$ then the eigenvectors associated to $a$ are $\lambda (1,0)^T$ where $\lambda\ne0$ and the eigenvectors associated to $-b$ are $\lambda(0,1)^T$ where $\lambda\ne0$.