We now take what we have learned about
eigenvalues and eigenvectors and relate
it back to diagonalizable matrices.  Now
what is a diagonalizable matrix?  Well, if
we have a square matrix A and there is a
nonsingular matrix X such that X
inverse A times X is equal to a diagonal
matrix then that matrix A is said to be
diagonalizable.  f you think about it
this is really just a discussion about
eigenvalues and eigenvectors in disguise.
How's that? Well let's have a look at it.
We can rewrite this as A times x is
equal to x times lambda. And then we
could say, "Well let's take our matrix X
and let's partition it by columns."  And we
do that on the left.  And we do that on
the right.  And then we can take our
matrix Lambda and we can expose its
diagonal elements.  Okay?
Now, multiplying out what we have on the
right here, just gives us the first
column is lambda_0 times x_0, and then
the next column would be lambda_1 times
x_1, and so forth, so here we get lambda_n-1 times x_n-1.  And on the
Left we know that multiplying a matrix
times the individual columns just means
multiplying that matrix times individual
columns.  And if we now set columns on the
left equal to columns on the right, what
we get is exactly the A times x_0 is
equal to lambda_0 times x_0 and so forth.
So from this, we conclude that if we have
such a matrix X then the columns of that matrix X equal
eigenvectors of our matrix A.  And the
corresponding diagonal elements of our
matrix Lambda equal the eigenvalues
corresponding to those eigenvectors.
Now we've already learned that, hey, it
may be that we don't actually have m
linearly independent eigenvectors.  So a
matrix is actually diagonalizable if and
only if it has m linearly independent
eigenvectors.  If we have m linearly
independent eigenvectors, we can make
them the columns of this matrix of this
matrix X and then we have exactly the X
that diagonalizes.  And if we have
the matrix X that diagonalizes the
matrix A, then we know that we can look
at its columns and find the eigenvectors of our matrix.  So the two go
hand-in-hand.
