One of the beauty of Matrix operation is using Eigen Value and Eigen Vector. Similar Matrices have the following property
Same Determinant
Same Rank
Same Eigen Value
If a matrix C is similar to matrix A than we can write
C = P^(-1) .A .P ( eq 1)
where P is an invertible matrix and C is a Diagonal matrix. Since A is also similar to C then
we can write
A = M^(-1).C.M (eq 2)
where M = P^(-1) which is easy to prove if one
Now the question is how to create a matrix P if we have a matrix A ?
Can we always find a Matrix Similar to matrix A ?
The diagonal matrix is one of the easiest one to operate.
The procedure to find the Diagonal matrix for a matrix A is to first find the eigen values of A. Now we know that corresponding to each eigen value we have at least one eigen vector.
Algebraic and Geometric Multiplicity
If we have a repeated eigen value then number of times that eigen value is repeated is called its algebraic multiplicity. The geometric multiplicity is the eigen vector space that eigen value gives. Thus geometric multiplicity tells us about the dimension of the eigen vector space corresponding to that eigen value.
There is a theorem which gaurantees that Geometric multiplicity of eigen space corresponding to a given eigen value cannot exceed its algebraic multiplicity. Which is interesting. So if we see an eigen vector having an algebraic multiplicity 1 we can be sure that its geometric multiplicity is also 1 and the eigen space is one dimensional.
Another theorem suggests if we have a n*n matrix and n independent eigen values then we have a n linearly independent eigen vectors also so we can take one vector from each eigen space and we can diagonalize the matrix. If however we have repeated eigen value than we have to be careful.The matrix will be diagonalizable only if for each eigen value we have same geometric multiplicity also. Each distinct eigen value gaurantees a distinct eigen vector. So if we have n independent eigen values then we can be sure that we can come up with n independent eigen vectors and these vectors in turn can form an n dimensional space.
Diagonalization is just creating a diagonal matrix whose diagonal elements are eigen values. Thus a diagonal form of a matrix depends on how we place our eigen values. There should be matching between eigen values and the way eigen vectors are placed. For example suppose we have eigen values p1,p2,p3 and eigen vectors v1, v2,v3 then the diagonal matrix
p2 0 0
0 p3 0
0 0 p1
will give the matrix as [v2 v3 v1]
similarly the matrix
p3 0 0
0 p1 0
0 0 p2
will require [v3 v1 v2]
So what does we mean by diagonalization of matrix A ?
Does it mean that similar matrices have same diagonalization ?
Yes it should be because if we go back to the similar matrices we C and A we defined we see that the matrices can be written in one form or the other and the only thing that change is the multiplication of P and M matrices.
What is the condition for a matrix to be orthogonally diagonalizable ?
Only Symmetric matrices are othogonally diagonalizable. If a matrix is not symmetric don't bother to find the orthogonally diagonalization.
