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The eigenvalues of the given matrix \(\begin{bmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{bmatrix}\) are:

- \(\lambda_1 = -2\)
- \(\lambda_2 = 3\)
- \(\lambda_3 = 6\)

The least eigenvalue is \(\lambda_1 = -2\). The corresponding eigenvector is:

\[
v = \begin{bmatrix} 0.7071 \\ -1.0419 \times 10^{-16} \\ -0.7071 \end{bmatrix}
\]

This can be approximated as \(v = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ -\frac{1}{\sqrt{2}} \end{bmatrix}\).

Would you like further details or clarifications? Here are some related questions to explore further:

1. How do we compute eigenvectors corresponding to eigenvalues?
2. Why do we normalize eigenvectors, and what does normalization mean?
3. Can a matrix have complex eigenvalues? If so, how do we find them?
4. What does an eigenvalue of 0 indicate about a matrix?
5. How can we interpret eigenvalues geometrically in the context of linear transformations?

**Tip:** The sum of the eigenvalues of a matrix equals the trace (sum of diagonal elements) of the matrix.

Find all the eigenvalues and eigenvector corresponding to least eigenvalue of the matrix [1 1 3; 1 5 1; 3 1 1].

Discover how to compute all the eigenvalues and the corresponding eigenvector for the least eigenvalue of a 3x3 matrix. This problem covers key concepts in linear algebra, including solving the characteristic equation and finding eigenvectors. Suitable for university students studying linear algebra.

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