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 ={{1,1,0},{1,-2,0}, {0,0,1}}

Eigenvalues[N[{{1,1,0},{1,-2,0},{0,0,1}}]]

1. Determine eigenvalues

Eigenvalues[N[M]]

2. Create eigenvectors

vecs = Eigenvectors[N[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] ]//MatrixForm

If we were calculating the principal stress directions and principal stress values
the eigenvalues would be the latter and the eigenvectors would be the former

Solving simlutaneous equations

Clear[a,b,c,d,e,f,g,h,i]
a =0;  b=1;c=0;d=1;e=-3;f=0;g=0;h=0;i=0;
{{a , b , c },{d ,e ,f },{g ,h , i }}.{x,y, z}=={0,0,0}
Solve[%,{x,y,z}]