D=(C^(-1))MC
	D is the diagonalized matrix. 
	M is the original matrix.
	 (C^(-1)) is the inverse matrix of C
	 C is the matrix of the transposed eigenvectors

Clear[M,vecs,A,DD]

M={{a,b},{c,d}}

Determine eigenvalues

Eigenvalues[M]

Create eigenvectors

vecs = Eigenvectors[M]

Transpose eigenvectors to create the C matrix

Transpose[vecs]

Inverse[Transpose[vecs]]

Full application of theorems to diagonalize a square matrix

DD=Simplify[Inverse[Transpose[vecs]].M.Transpose[vecs] ]

Now, let’s perform diagonalization on a matrix of numbers.
Recall from above, that
D=(C^(-1))MC, where
	D is the diagonalized matrix. 
	M is the original matrix.
	(C^(-1)) is the inverse matrix of C
	C is the matrix of the transposed eigenvectors

M ={{1,2},{2,-5}}

1. Determine eigenvalues

Eigenvalues[M]

2. Create eigenvectors

vecs = Eigenvectors[M]

Transpose eigenvectors to create the C matrix

Transpose[vecs]

Inverse[Transpose[vecs]]

Full application of theorems to diagonalize a square matrix

DD=Simplify[Inverse[Transpose[vecs]].M.Transpose[vecs] ]