Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}2 & 1\\-1 & 2\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}2 & 1\\-1 & 2\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}2 - \lambda & 1\\-1 & 2 - \lambda\end{array}\right]$$$.

The determinant of the obtained matrix is $$$\lambda^{2} - 4 \lambda + 5$$$ (for steps, see determinant calculator).

Solve the equation $$$\lambda^{2} - 4 \lambda + 5 = 0$$$.

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

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = 2 - i$$$

  $$$\left[\begin{array}{cc}2 - \lambda & 1\\-1 & 2 - \lambda\end{array}\right] = \left[\begin{array}{cc}i & 1\\-1 & i\end{array}\right]$$$

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

  This is the eigenvector.

• $$$\lambda = 2 + i$$$

  $$$\left[\begin{array}{cc}2 - \lambda & 1\\-1 & 2 - \lambda\end{array}\right] = \left[\begin{array}{cc}- i & 1\\-1 & - i\end{array}\right]$$$

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

  This is the eigenvector.

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

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