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:

[1423]I found the eigenvalues of this matrix are -1 and 5, but what are the multiplicities of these?

**Answer**

The characteristic polynomial of the matrix is pA(x)=det. In your case, A = \begin{bmatrix} 1 & 4 \\ 2 & 3\end{bmatrix}, so p_A(x) = (x+1)(x-5). Hence it has two distinct eigenvalues and each occurs only once, so the algebraic multiplicity of both is one.

If B=\begin{bmatrix} 5 & 0 \\ 0 & 5\end{bmatrix}, then p_B(x) = (x-5)^2, hence the eigenvalue 5 has algebraic multiplicity 2. Since \dim \ker (5I-B) = 2, the geometric multiplicity is also 2.

If C=\begin{bmatrix} 5 &1 \\ 0 & 5\end{bmatrix}, then p_C(x) = (x-5)^2 (same as p_C), hence the eigenvalue 5 has algebraic multiplicity 2. However, \dim \ker (5I-C) = 1, the geometric multiplicity is 1.

**Very** loosely speaking, the matrix is ‘deficient’ in some sense when the two multiplicities do not match.

The algebraic multiplicity of an eigenvalue \lambda is the power m of the term (x-\lambda)^m in the characteristic polynomial.

The geometric multiplicity is the number of linearly independent eigenvectors you can find for an eigenvalue.

**Attribution***Source : Link , Question Author : Johnathon Svenkat , Answer Author : le_m*