Eigenvalues and eigenvectors of $$$\left[\begin{array}{c}e^{a}\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{c}e^{a}\end{array}\right]$$$.

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

The determinant of the obtained matrix is $$$- \lambda + e^{a}$$$ (for steps, see determinant calculator).

Solve the equation $$$- \lambda + e^{a} = 0$$$.

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

These are the eigenvalues.

Next, find the eigenvectors.

$$$\lambda = e^{a}$$$

$$$\left[\begin{array}{c}- \lambda + e^{a}\end{array}\right] = \left[\begin{array}{c}0\end{array}\right]$$$

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

This is the eigenvector.

Eigenvalue: $$$e^{a}$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}1\end{array}\right]$$$A.