###### Example4.2.2

If a matrix has the characteristic equation

\begin{equation*} (4-\lambda)^2(-5-\lambda)(1-\lambda)^7(3-\lambda)^2 = 0\text{,} \end{equation*}

then that matrix has four eigenvalues: $$\lambda=4$$ having multiplicity 2; $$\lambda=-5$$ having multiplicity 1; $$\lambda=1$$ having multiplicty 7; and $$\lambda=3$$ having multiplicty 2. The degree of the characteristic polynomial is the sum of the multiplicities $$2+1+7+2 = 12$$ so this matrix must be a $$12\times12$$ matrix.

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