, what wrong with considering Spectral form of http://en.wikipedia.org/wiki/Spectral_theorem#Normal_matrices"?A symmetric matrix is a square matrix in which the elements above and below the main diagonal are reflections of each other. In other words, for a matrix A, if A[i,j] = A[j,i] for all i and j, then A is a symmetric matrix.
A normal matrix is a square matrix that commutes with its conjugate transpose. In other words, for a matrix A, if A * A* = A* * A, where A* is the conjugate transpose of A, then A is a normal matrix.
To show that a matrix is symmetric, you need to prove that A[i,j] = A[j,i] for all i and j. This can be done by either manually checking each element or by using the property of symmetry that the transpose of a matrix is equal to itself.
To show that a matrix is normal, you need to prove that A * A* = A* * A. This can be done by multiplying the matrix A and its conjugate transpose A* and checking if the result is equal to the conjugate transpose of A multiplied by A.
Symmetric and normal matrices have important properties that make them useful in various applications in linear algebra. For example, symmetric matrices have real eigenvalues and can be easily diagonalized, while normal matrices can be diagonalized by unitary matrices. These properties make them valuable in solving systems of linear equations, performing transformations, and other operations.