# Characteristic polynomial of a $3\times 3$ matrix

Let $$M = \begin{bmatrix} 1 &1 & 3 \\[0.3em] 1 & 5 & 1 \\[0.3em] 3 & 1 & 1 \end{bmatrix}$$ a matrix. I am stuck on solving this question from my book. Find eigenvalues, characteristic polynomial and diagonalize this matrix.

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Characteristic polynomial = $det(A-\lambda I)$ –  HipsterMathematician Nov 10 '12 at 16:18
What have you tried? –  Chris Eagle Nov 10 '12 at 16:23
@charlie. Now I can find characteristic P. can you help on other requests of question. –  Milingona Ana Nov 10 '12 at 16:37
@MilingonaAna Did you see my answer? –  HipsterMathematician Nov 10 '12 at 17:09
$Charlie. My question is not as easy as it seems. Someone (not you) hurried up and gave me negative vote. For you +1 from me. – Milingona Ana Nov 10 '12 at 17:18 ## 1 Answer You have that the characteristic polynomial is given by:$\det(M-\lambda I), so we have: $$M = \begin{bmatrix} 1-\lambda &1 & 3 \\[0.3em] 1 & 5-\lambda & 1 \\[0.3em] 3 & 1 & 1-\lambda \end{bmatrix}$$ Find the determinant. \begin{align*} \det(M-\lambda I) &=(1-\lambda)(5-\lambda)(1-\lambda)+3+3-9(5-\lambda)-(1-\lambda)-(1-\lambda) \\ &=(1-\lambda)(5-\lambda)(1-\lambda)-2(1-\lambda)+6. \end{align*} This is the characteristic polynomial. To find the eingenvalues, make it equal to zero then find the roots of such polynomial.Then you're done. Now, for the diagonal matrix :M$is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix$P$such that$P^{−1}MP$is a diagonal matrix. You have to find the eigenvector related to the eigenvalues you found.Suppose they are$\lambda _1,\lambda _2, \lambda _3.$They will appear in the diagonal of the diagonal matrix you look for.The proccess to find such matrix is the following: 1. You find the eigenvectors related to the eigenvalues you found; 2. Suppose they are: $$v_1=[a_1, a_2, a_3],\ v_2=[b_1, b_2, b_3],\ v_3=[c_1, c_2,c_3].$$ Now let$P$be the matrix of these eigenvectors as its columns: $$\begin{bmatrix} a_1 &b_1 & c_1 \\[0.3em] a_2 & b_2 & c_2 \\[0.3em] a_3 & b_3 & c_3 \end{bmatrix}$$ Now you will have to find$P^{-1}$.Once you found it you have to compute:$P^{-1}MP\$, which is equal to your diagonal matrix with the eigenvalues in the main diagonal.It's a good exercise to check this way.

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how to diagonalize –  Adi Dani Nov 10 '12 at 16:44
First, you have to check if it is diagonalizable –  HipsterMathematician Nov 10 '12 at 16:46