Let A be an n-by-n matrix with complex entries.
And let lambda be an eigenvalue of A.
We're now going to look at two
quantities associated with lambda.
The first is called the algebraic
multiplicity of lambda.
Remember that the eigenvalues are precisely
the roots of the characteristic polynomial of A.
And the algebraic multiplicity of lambda
is the number of times lambda
appears as a root of the characteristic polynomial.
For example, if my matrix A
is this, then the characteristic
polynomial of A is given by
1 minus lambda, times 2 minus
lambda squared.  And so there are 
two distinct eigenvalues, 2 and 1.
And 2 will have algebraic multiplicity 2
because it appears as a root of the
characteristic polynomial two times,
whereas 1 will have algebraic multiplicity 1.
The other quantity is called
the geometric multiplicity.
The geometric multiplicity is given by
the dimension of the eigenspace for the
eigenvalue lambda.  And the eigenspace for the eigenvalue lamda
will be denoted by script E subscript A
open bracket, lambda, close bracket.
And we're now going to look at the
definition of eigenspace.
So the eigenspace of A for lambda
is given by the nullspace of the matrix
A minus lambda times the identity matrix.
Remember that the nonzero vectors
in the nullspace of A minus lambda I
are precisely the eigenvectors of A
with eigenvalue lambda.
So let's go back to the matrix here and
see what the geometric multiplicity
of the eigenvalue 2 is.
So we're looking at the nullspace of A - 2 I
which is precisely 0 1 0,
0 0 0, and 0 0 -1.
And this matrix is rank 2.
So the dimension
of the nullspace of A - 2 I is 1.
So what that means is
the eigenvalue 2 has geometric multiplicity 1.
Notice that for the eigenvalue 2,
the geometric multiplicity is
less than the algebraic multiplicity.
And in general,
the geometric multiplicity of any eigenvalue
cannot exceed the algebraic multiplicity
of the eigenvalue. and we have seen
that
And we have seen that
in general, this inequality can be strict.
Now notice that if you an eigenvalue that 
has algebraic multiplicity 1,
then automatically its geometric
multiplicity must be 1 as well.
First of all, the geometric multiplicity of
that eigenvalue cannot exceed 1 in this
case, but it has to be at least one
because by definition
we must have a nonzero vector in the nullspace
of A minus lambda I.
And that means the rank of the matrix 
cannot be full.
So the dimension of the nullspace
must be at least 1.
We'll see later on that
the geometric multiplicities of the eigenvalues
can help us determine if we can 
diagonalize a matrix.
Diagonalizing a matrix will help us
compute powers of the matrix.
And that's something we'll look at 
in more detail in a later video.
