Category: proofs

The eigenvalues of a square matrix with real entries can be complex. For example, a quick calculuation will verify that the eigenvalues of $$A = \begin{pmatrix} 1&-2\\2&1\end{pmatrix}$$ are $1\pm2i$. But when the matrix is symmetric, the eigenvalues must be real.…Continue readingEigenvalues of Real Symmetric Matrices

In this post we will give a proof of the following theorem. Theorem If $\vec v_1, \vec v_2, \ldots , \vec v_r$ are eigenvectors of matrix $A$ that correspond to distinct eigenvalues $\lambda_1, \lambda_2, \ldots , \lambda_r$, then the set…Continue readingWhy are Eigenvectors from Distinct Eigenvalues Linearly Independent?