Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & 0\\0 & 1\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}1 - \lambda & 0\\0 & 1 - \lambda\end{array}\right]$$$.

The determinant of the obtained matrix is $$$\left(1 - \lambda\right)^{2}$$$ (for steps, see determinant calculator).

Solve the equation $$$\left(1 - \lambda\right)^{2} = 0$$$.

The roots are $$$\lambda_{1} = 1$$$, $$$\lambda_{2} = 1$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

$$$\lambda = 1$$$

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

The null space of this matrix is $$$\left\{\left[\begin{array}{c}1\\0\end{array}\right], \left[\begin{array}{c}0\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

These are the eigenvectors.

Eigenvalue: $$$1$$$A, multiplicity: $$$2$$$A, eigenvectors: $$$\left[\begin{array}{c}1\\0\end{array}\right]$$$, $$$\left[\begin{array}{c}0\\1\end{array}\right]$$$A.