Question

The following are the Jordan Canonical Form of linear transformations. Find the characteristic polynomial, minimal polynomials, the algebraic multiplicity, geometric multiplicity and the index of each of the eigenvalues of L.

A=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)

Answer 1

Solution

A=\( \begin{pmatrix}2&0&0&0\\ 0&2&0&0\\ 0&0&1&0\\ 0&0&0&3\end{pmatrix} \)

The characteristics polynomial. \( \implies P(\lambda)=|A-\lambda I|=\bigg(2-\lambda\bigg)^2\bigg(1-\lambda\bigg)\bigg(3-\lambda\bigg)\hspace{2mm} \)

The algebraic multiplicity

\( \implies am(2)=2,am(1)=1,am(3)=1 \hspace{2mm} \)

The geometric multiplicity

\( \implies gm(2)=2,gm(1)=1,gm(3)=1\hspace{2mm} \)

The \( index\implies Index(2)=1,Index(1)=1,Index(3)=1 \hspace{2mm} \)

The minimal polynomial

\( \implies m(\lambda)=\bigg(\lambda-1\bigg)\bigg(\lambda-1\bigg)\bigg(\lambda-3\bigg)\hspace{2mm} \)

Answer :

Therefore .

\( \implies P(\lambda)=|A-\lambda I|=\bigg(2-\lambda\bigg)^2\bigg(1-\lambda\bigg)\bigg(3-\lambda\bigg)\hspace{2mm} \)

\( \implies am(2)=2,am(1)=1,am(3)=1 \hspace{2mm} \)

\( \implies gm(2)=2,gm(1)=1,gm(3)=1\hspace{2mm} \)

\( \implies Index(2)=1,Index(1)=1,Index(3)=1 \hspace{2mm} \)