Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & 3\\1 & -1\end{array}\right]$$$
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Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}1 & 3\\1 & -1\end{array}\right]$$$.
Solution
Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}1 - \lambda & 3\\1 & - \lambda - 1\end{array}\right]$$$.
The determinant of the obtained matrix is $$$\lambda^{2} - 4$$$ (for steps, see determinant calculator).
Solve the equation $$$\lambda^{2} - 4 = 0$$$.
The roots are $$$\lambda_{1} = -2$$$, $$$\lambda_{2} = 2$$$ (for steps, see equation solver).
These are the eigenvalues.
Next, find the eigenvectors.
$$$\lambda = -2$$$
$$$\left[\begin{array}{cc}1 - \lambda & 3\\1 & - \lambda - 1\end{array}\right] = \left[\begin{array}{cc}3 & 3\\1 & 1\end{array}\right]$$$
The null space of this matrix is $$$\left\{\left[\begin{array}{c}-1\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).
This is the eigenvector.
$$$\lambda = 2$$$
$$$\left[\begin{array}{cc}1 - \lambda & 3\\1 & - \lambda - 1\end{array}\right] = \left[\begin{array}{cc}-1 & 3\\1 & -3\end{array}\right]$$$
The null space of this matrix is $$$\left\{\left[\begin{array}{c}3\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).
This is the eigenvector.
Answer
Eigenvalue: $$$-2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}-1\\1\end{array}\right]$$$A.
Eigenvalue: $$$2$$$A, multiplicity: $$$1$$$A, eigenvector: $$$\left[\begin{array}{c}3\\1\end{array}\right]$$$A.