# Can a matrix have eigenvalue with infinite multiplicity?

Suppose we have matrix of the form

$$A= \begin{bmatrix} a & -1 \\ 0 & a \\ \end{bmatrix}$$

and we would like to analyze its diagonalizability.

By taking the characteristic equation we quickly find $a^2 = 0$. In this case, can we say that this matrix has eigenvalue of 0 with infinite multiplicity?

• The characteristic equation is $x \mapsto (x-a)^2$ which shows that $a$ is an eigenvalue of multiplicity two. Where do you get the infinite from? – copper.hat Jan 27 '15 at 20:17
• And no, an eigenvalue is a zero of a polynomial, so it can only have finite multiplicity. – copper.hat Jan 27 '15 at 20:19
• How did you decide that $a^2 = 0$? – Ben Grossmann Jan 27 '15 at 21:06

## 2 Answers

You're computing the determinant of $A$, which is not the characteristic polynomial. The characteristic polynomial is rather $$\det(A-XI)=\det\begin{bmatrix}a-X&-1\\0&a-X\end{bmatrix}=(a-X)^2$$ which has one root (equal to $a$) with multiplicity $2$.

The matrix is not diagonalizable, because $$A-aI=\begin{bmatrix}0&-1\\0&0\end{bmatrix}$$ has rank $1$.

Nope. This is a polynomial equation in $a$ of degree $2$, so that if there is only one $a\in\mathbb R$ with $a^2=0$ (and this is the case: any $a\neq 0$ will have $a^2>0$), then this $a$ will be a double root, that is, a root of multiplicity $2$.

In the end, you can never have an eigenvalue of multiplicity greater than $n$ for an $n\times n$ matrix, because the characteristic equation of an $n\times n$ matrix is always a polynomial equation of degree $n$.