1. Similar diagonalization of real symmetric matrix
I.Properties of eigenvalues and eigenvectors of real symmetric matrix.
Property 1: the eigenvalues of real symmetric matrix are all real.
(1)transpose each side, then
Property2: The eigenvectors which corresponding to unequal eigenvalues of real symmetric matrix are orthogonal.
For general matrices, we only know that the
eigenvectors of unequal eigenvalues are linearly independent.

3. Property3:There are k linearly independent eigenvectors
of k-multiple eigenvalues of real symmetric matrix A.
So, real symmetric matrix A can be diagonalized.
II. Similar diagonalization of real symmetric matrix
Theory1:real symmetric matrix must be similar to a diagonal matrix.
Theory2: real symmetric matrix A must be similarly orthogonal to a
diagonal matrix.

4. And we get

5. Process of diagonalizing real symmetric
matrix A by orthogonal matrices.

9. complementarity

12. III. Congruent matrices~ .
Congruent matrices are reflexive, symmetric, transitive.
The relation of equivalence, similarity, congruence: ~
And the inverse is wrong.
Generally, similarity and
congruence have no relation.
And orthogonal similarity equals to congruence.
Theory：A real symmetric matrix must be congruent
to a diagonal matrix.

13. Application of similar matrices

14. = 0