Why do real symmetric matrices have to be diagonalized?

The real symmetric matrix must be diagonalized, because the necessary and sufficient condition for similar diagonalization is that the n-order square matrix A has n linearly independent eigenvectors, and the sufficient condition is that A has n different eigenvalues, and the n different eigenvalues must correspond to n linearly independent eigenvectors, so the real symmetric matrix must be diagonalized.

Extended data:

Properties of real symmetric matrices;

1 and the eigenvectors corresponding to the different eigenvalues of the real symmetric matrix A are orthogonal.

2. The eigenvalues of real symmetric matrix A are all real numbers, and the eigenvectors are all real vectors.

3. The real symmetric matrix A of order n must be diagonalizable, and the elements on the similar diagonal matrix are the eigenvalues of the matrix itself.

4. If λ0 has k multiple eigenvalues, then there must be k linearly independent eigenvectors, or there must be a rank r(λ0E-A)=n-k, where e is identity matrix.