Example: 2x2 and 4x4 matrix

2.13.2 Example: 2x2 and 4x4 matrix

\[ \tilde A=\left(\begin{array}{cc}1&-2\\-2&1\end{array}\right)\;\;2\times2\]

\begin{eqnarray*} \det(\tilde A-\lambda\tilde I)&=&\left|\begin{array}{cc}1-\lambda&-2\\-2&1-\lambda\end{array}\right|=(1-\lambda)^2-4=0\\\\ \det(\tilde A-\lambda\tilde I)&=&P(\lambda)=\lambda^2-2\lambda-3=0\\\\ P(\lambda)&=&\lambda^2-2\lambda-3\;\mbox{is the characteristic polynomial associated with the matrix } \tilde A \quad.\\ P(\lambda)&=&0\;\rightarrow\;\lambda_{1/2}=1\pm\sqrt{1^2+3}=1\pm2\rightarrow\begin{array}{c}\lambda_1=3\\\lambda_2=-1\end{array}\\ &\Rightarrow&\;\mbox{Eigenvalues of matrix $\tilde A$ are $\lambda_1=3$ and $\lambda_2=-1$}\end{eqnarray*}
\[\tilde{A}=\left(\begin{array}{cccc}1&0&0&1\\ 0&1&0&0\\ 0&0&1&0\\ 1&0&0&1\end{array}\right)\;4\times4\]

\begin{eqnarray*} \det(\tilde{A}-\lambda\tilde{I})&=&\left|\begin{array}{cccc}1-\lambda&0&0&1\\0&1-\lambda&0&0\\0&0&1-\lambda&0\\ 1&0&0&1-\lambda \end{array}\right|=(1-\lambda)\left|\begin{array}{ccc}1-\lambda&0&1\\ 0&1-\lambda&0\\ 1&0&1-\lambda \end{array}\right|\\\\ &=&(1-\lambda)^2\left|\begin{array}{c}1-\lambda\\1\end{array}\begin{array}{c}1\\1-\lambda\end{array}\right|\\\\ &=&(1-\lambda)^2\left[(1-\lambda)^2-1\right]=(1-\lambda)^2\left(\lambda^2-2\lambda+1-1\right)\\\\ \Rightarrow\;P(\lambda)&=&(1-\lambda)^2\lambda(\lambda-2)\;\left(\mbox{important: not further simplifying!!}\right) \end{eqnarray*}
Eigenvalues:

\[\begin{array}{lccl}P(\lambda)=0&\rightarrow&\lambda_1=1&(\rightarrow\;\mbox{two times because of } (1-\lambda)^2)\\&&\lambda_2=0\\ &&\lambda_3=2\end{array}\]

Matrix $\tilde A,\;N\times N\;\Rightarrow\;$ Polynomial $P(\lambda)$ is of degree $N$; $N$ Eigenvalues exist, since $P(\lambda)=0$ has $N$ solutions
However, example (ii) only yields 3 EW for a $4\times4$ matrix because $\lambda_1=1$ is a multiple zero.
in general $\lambda_0$ is called an $j$-times zero (zero of order $j$) of $P(\lambda)$ if

\[P(\lambda)=(\lambda-\lambda_0)^jP_0(\lambda)\quad\mbox{and}\quad P_0(\lambda)\;\mbox{is of degree $N-j$}\]

Example:

\begin{eqnarray*}P(\lambda)&=&(\lambda-1)^2\lambda(\lambda-2)\\ &&\left.\begin{array}{lcl}\lambda_1=1&\rightarrow&\mbox{2 times}\\\lambda_2=0&\rightarrow&\mbox{1 times}\\\lambda_3=2&\rightarrow&\mbox{1 times}\end{array}\right\}\;\mbox{4 zeros since $\lambda_1$ counts twice!}\end{eqnarray*}
\begin{eqnarray*}\left(\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right)\;\rightarrow\;&P(\lambda)&=(1-\lambda)^4=(\lambda-1)^4\\ &\rightarrow&\lambda_1=1\;\mbox{is a 4 times zero of $P(\lambda)$}\end{eqnarray*}

With frame

Eigenvalues and Eigenvectors