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

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

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

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

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

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

These are the eigenvalues.

Next, find the eigenvectors.

$$$\lambda = t$$$

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

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

This is the eigenvector.

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