## Similar Matrices, Diagonalizable

### Diagonalizable

A matrix M is diagonalizable if it is similar to a diagonal matrix.
Don't confuse this with the process of converting a matrix into
row-echelon form.
Both processes create a diagonal matrix,
but they are otherwise unrelated.
Once again, M is diagonalizable if PM/P = D,
where P is nonsingular and D is diagonal.

Let's look at the eigen vectors of D.
Take a unit coordinate vector and multiply by D to get a scaled version of that vector.
Thus D has n independent eigen vectors, one for each coordinate.

In the last section we showed that eigen vectors in one matrix
map onto the eigen vectors of a similar matrix,
and this map preserves linear independence.
Therefore every diagonalizable matrix
(which is similar to a diagonal matrix)
has n independent eigen vectors.

Conversely, let M have n independent eigen vectors.
Let P map the unit basis onto these n independent vectors.
Apply M, which scales these vectors, then apply P inverse,
which pulls the vectors back to the coordinate axes.
The composite is a matrix that scales the coordinate vectors,
in other words, a diagonal matrix.
Therefore a matrix is diagonalizable iff it has n independent eigen vectors.

An example of a matrix with too few eigen vectors is [1,1|0,1].
It has one eigen value, 1, and one eigen vector, the y axis.
It is not similar to a diagonal matrix,
i.e. not diagonalizable.

If D is the identity matrix, or a scale multiple of the identity matrix, it commutes with P.
Write PM = DP = PD and cancel P on the left to get M = D.
Only the identity matrix is similar to itself.