Diagonalize $\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]$

Diagonalize $\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]$.

First, find the eigenvalues and eigenvectors (for steps, see eigenvalues and eigenvectors calculator).

Eigenvalue: $1$, eigenvector: $\left[\begin{array}{c}5\\1\end{array}\right]$.

Eigenvalue: $-2$, eigenvector: $\left[\begin{array}{c}2\\1\end{array}\right]$.

Form the matrix $P$, whose column $i$ is eigenvector no. $i$: $P = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]$.

Form the diagonal matrix $D$ whose element at row $i$, column $i$ is eigenvalue no. $i$: $D = \left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]$.

The matrices $P$ and $D$ are such that the initial matrix $\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right] = P D P^{-1}$.

$P = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]$A

$D = \left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]$A