$$
A= \begin{pmatrix} -3 & 4\\ 4 & 3
\end{pmatrix}
$$
So i am assuming that i must find the evalues and evectors of this matrix first, and that is exactly what i did.
The evalues are $5$ and $-5$, and the evectors are $(2,1)^T$ and $(1,-2)^T$
Now the spectral decomposition of $A$ is equal to $(Q^{-1})^\ast$ (diagonal matrix with corresponding eigenvalues) * Q
$Q$ is given by [evector1/||evector1|| , evector2/||evector2||]
and for Q i got the matrix
$$
Q= \begin{pmatrix} 2/\sqrt{5} &1/\sqrt{5} \\ 1/\sqrt{5} & -2/\sqrt{5}
\end{pmatrix}
$$
the inverse of Q is the matrix…
$$
\begin{pmatrix} 2 \sqrt{5}/5 & \sqrt{5}/5 \\ \sqrt{5}/5 & -2 \sqrt{5}/5
\end{pmatrix}
$$
and the diagonal matrix with corresponding evalues is
$$
A= \begin{pmatrix} 5 & 0\\ 0 & -5
\end{pmatrix}
$$
so now i found the spectral decomposition of $A$, but i really need someone to check my work.
Did i take the proper steps to get the right answer, did i make a mistake somewhere?
Best Answer
The needed computation is
$$\mathsf{A} = \mathsf{Q\Lambda}\mathsf{Q}^{-1}$$
Where $\Lambda$ is the eigenvalues matrix. And your eigenvalues are correct.
Hence you have to compute
$$\mathsf{AQ} = \mathsf{Q\Lambda}$$
Which gives you the solutions
$$a = 2c ~~~~~~~~~~~ d = -\frac{b}{2}$$
You can then choose easy values like $c = b = 1$ to get
$$Q = \begin{pmatrix} 2 & 1 \\ 1 & -\frac{1}{2} \end{pmatrix}$$
And easily
$$\mathsf{Q}^{-1} = \frac{1}{\text{det}\ \mathsf{Q}} \begin{pmatrix} -\frac{1}{2} & -1 \\ -1 & 2 \end{pmatrix}$$
Which you can compute alone.