# Thread: Finding a basis

1. ## Finding a basis

Find basis for R3 relative to which the matrix for T is diagonal:

T[x1, x 2, x3] = [-2x1 + x2 - x3, x1 - 2x2 - x3, -x1 - x2 - 2x3]

2. Hello

Originally Posted by chadlyter
Find basis for R3 relative to which the matrix for T is diagonal:

T[x1, x 2, x3] = [-2x1 + x2 - x3, x1 - 2x2 - x3, -x1 - x2 - 2x3]
Find the matrix T of the transformation.

$T=\begin{pmatrix} -2&1&-1 \\ 1&-2&-1 \\ -1&-1&-2 \end{pmatrix}$

Step 1 :
find the eigenvalues

Step 2 :
find the eigenvectors associated to each eigenvalue

Step 3 :
it's done :P

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How to find eigenvalues :

Find the characteristic polynomial, defined as :

$\chi_T(\lambda)=det(T-\lambda I_3)$

The eigenvalues will be the roots of the characteristic polynomial.
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Comment :
Hmmm I have to check it out