A symmetric matrix $A$ is always of the form
$$ A=V^t D V $$where $V$ is a orthogonal matrix and $D$ is a diagonal one. This is a version of the spectral theorem.
The meaning is that symmetric matrices correspond to scale changes. While diagonal matrices encode scale changes in the main axis, a general symmetric matrix represents a scale change in another (orthogonal) axis, and $V$ is the orthogonal transformation between these axis. In a sense, they are a kind of generalization of diagonal matrices.
Symmetric matrices are, therefore, always diagonalizable and have orthogonal eigenvectors. This is generalized to self adjoint operator in finite dimensional Hilbert spaces or, better said, to self adjoint matrixs
Keep an eye: is not the same as symmetric operator, but is related.
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Author of the notes: Antonio J. Pan-Collantes
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