TITLE: How to find the multiplicity of eigenvalues?
QUESTION [59 upvotes]: I don't understand how to find the multiplicity for an eigenvalue. To be honest, I am not sure what the books means by multiplicity. 
For instance, finding the multiplicty of each eigenvalue for the given matrix:
$$\begin{bmatrix}1 & 4\\2 & 3\end{bmatrix}$$
I found the eigenvalues of this matrix are -1 and 5, but what are the multiplicities of these?

REPLY [14 votes]: Let me explain the two multiplicities that I know are related to eigen-values of matrices:
Firstly, what is the eigenvalue of a matrix $A$? By definition it consists of the zeros of the polynomial: $\det(A-xI)$. So the muliplicities that they occur in this polynomial are defined to be the multiplicities of the eigen-values.
Secondly, since, for an eigen-value $\lambda$, we have $\det(A-\lambda I)=0$, i.e. $A-\lambda I$ is a singular matrix, and the linear transformation it defines has a non-trivial kernel. The dimension of this kernel is then said to be the geometric multiplicity of the eigen-value.
Hence, in one case, one has to compute some polynomial; while, on the other hand, one has to compute some transformations, to find its kernel, and to determine the dimension of the kernel, to find the multiplicites of eigen-values. Notice here that you have a $2\times 2$ matrix with $2$ eigen-values, and hence they must be of multiplicities $1$. That is to say, the two notions coincide, and the matrix in question must be diagonalisable.