# How do I show that a given lambda value is an eigen value of a matrix?

Also how do I show that a matrix is not diagonalizable based on my calculations from the equation $\det(A-\lambda I)=0$

For the first part say the matrix is:

$$\begin{pmatrix} 1 & -2 & 3 \\ 0 & -1 & 3 \\ -1 & 2 & -2 \\ \end{pmatrix}$$ is $\lambda=1$ an eigenvalue for the matrix?

Also I wouldn't mind finding out the code for lambda on here and the code for putting a set vectors in brackets as related to span.

Thanks!

• Code for lambda is "\lambda" between dollar signs. – Dave May 1 '16 at 20:43
• Also, the curly brackets { and } have to be "escaped" in MathJax/$\LaTeX$ markup by preceding them with backslash \. See the MathJax Basic Tutorial and Quick Reference for many of the common techniques. – hardmath May 19 '16 at 12:16

To find the eigenvalues of $A=\begin{bmatrix} 1&-2&3\\0&-1&3\\-1&2&-2\end{bmatrix}$ we compute the roots of the characteristic polynomial, $p_A(t)=det(tI-A)$. Using Laplace Expansion we have: $$p_A(t)=det\begin{bmatrix} t-1&2&-3\\0&t+1&-3\\1&-2&t+2\end{bmatrix}=(t-1)det\begin{bmatrix} t+1&-3\\-2&t+2 \end{bmatrix}+det\begin{bmatrix} 2&-3\\t+1&-3 \end{bmatrix}=(t-1)[(t+1)(t+2)-6]+[-6+3t+3]=(t-1)(t^2+3t-1)$$ From factored form, we see that the eigenvalues of $A$ are $\lambda_1=1, \lambda_2=\frac{-3+\sqrt {13}}{2}, \lambda_3=\frac{-3-\sqrt {13}}{2}$. So to answer the first question: yes, the matrix has $1$ as an eigenvalue. Now, because the matrix is $3\times 3$ and it has $3$ distinct eigenvalues, it is diagonalizable.
Note that showing an $n\times n$ matrix is diagonalizable is equivalent to showing that the algebraic multiplicities of its eigenvalues equals their geometric multiplicities. In the special case where it has $n$ distinct eigenvalues, this condition is immediately seen to hold true, and the matrix is diagonalizable.
After subtracting the eigenvalues $\lambda$ from diagonal elements are some rows or columns or after some linear combinations the same? Does it vanish?
$$\begin{pmatrix} 0 & -2 & 3 \\ 0 & -2 & 3 \\ -1 & 2 & -3 \\ \end{pmatrix}$$