Matrix, diagonalizable, eigenvalues

I never quite understand what matrix is diagonalizable, what are the relation of eigenvalues with special matrix, until now. The main goal here is to understand real square matrix, but occasionally we will need to refer to complex matrix as well.

1. Real Invertible Matrix iff eigenvalues are in C\{0}. Nonzero eigenvalues
2. Hermitian matrix, and special case symmetric real matrix implies real eigenvalues. All real eigenvalues does not imply symmetric. Counter example: [[0,1],[4,0]] has eigenvalues 2 and -2, and is not symmetric. More later on what can be implied from all real eigenvalues. 
3. A "positive definite" matrix, i.e. x'Ax>0 for all x not 0, has all eigenvalues with positive real part. A symmetric/Hermitian and positive definite matrix implies eigenvalues are real and positive, since symmetric/Hermitian implies the real, and positive definite implies the positive real part.   Positive eigenvalues and symmetric/Hermitian imply positive definite. Positive eigenvalues alone does not imply positive definite. E.g. [[1,-3],[0,1]] has double eigenvalues 1, but for v = [1,1], the quadratic form is negative. 
4. A skew-symmetric real matrix have eigenvalues with real part equal to zero. 
5. In the complex setting, a matrix is unitary diagonalizable iff it is a normal matrix AA*=A*A. It is a two way relation. Spectral Theorem. 
6. In the real setting, a matrix is orthogonal diagonalizable iff it is symmetric. Again it is a two way relation. One way is implied from Spectral. The other is easy by definition. 
7. In the real setting, a matrix is diagonalizable (i.e. A = PDP^{-1} for P invertible, no need for orthonormal as above) iff it has all real eigenvalues, and its minimal polynomial has no repeated root. Having all real eigenvalues alone is not enough, for example: [[1,1,0],[0,1,0],0,0,1]. This is in Jordan Canonical Form, so can not be diagonalized further. In fact the eigenvalues is all one, and its has only two eigenvectors (1,0,0) and (0,0,1). The eigenspace have dimension 2. 
8. In the complex setting, a matrix is diagonalizable again iff its minimal polynomial has no repeated root (complex already contains all root so no need for first condition). 

Finally, for any general n*p matrix A, A = UDV, for U n*n, V p*p are orthogonal matrix, D n*p matrix with r positive elements down the diagonal, every other element zero. r is the rank of A, A', A'A, AA'.