For the $3\times 3$ matrix $A$ with eigenvalues $x_1$, $x_2$, $x_3$ find idempotents matrix $E_1,E_2.E_3$ so that $A=x_1E_1+x_2E_2+x_3E_3$ .

Question

Let $A=\begin{pmatrix} 1 & -14 & -10 \\ -14 & -2 & -4 \\ -10 & -4 & 10 \\ \end{pmatrix}$. Find the eigenvalues of $A$ and idempotents matrix $E_1,E_2.E_3$ so that $A=x_1E_1+x_2E_2+x_3E_3$.

The matrix $A$ have eigenvalues $x_1=9$, $x_2=-18$ and $x_3=18$.

I don't have idea of how build the idempotents matrix $E_1,E_2.E_3$. I wait that you can give me a hints or say me the theorem for find this idempotents matrix.

Answer 1

For each eigenvalue, find an eigenvector, call it $w,$ as a column vector. Next, define column vector $v = w /|w|,$ which is now a unit length vector. You should confirm unit length.

Now, if you write $v^T v$ you just get the one by one matrix with single entry $1;$ usually people just say $v^T v = 1.$

Instead, the idempotent matrix for that eigenvalue is $$ E = v v^T $$ This is a symmetric matrix of rank one, positive semi-definite but not definite. It has trace exactly $1.$ Finally, $E^2 = v v^T v v^T = v (v^T v) v^T = v (1) v^T = v v^T = E$