How to find the multiplicity of eigenvalues?

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:
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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

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