Let be linear transformations on an
-dimensional vector space such that
and
for
. Then
.
How does this happen? Take the expression and multiply by any
on both sides. We see that
. Hence any vector
can be expressed as a sum of elements in
for
.
Why do we have a direct sum decomposition? Let for
. Then consider
. For any
where
, $v_i=E_i v$ for some
. Hence
. Hence, we have
. Now
for some
. Hence
. Note that
(just multiply the expression
by
on both sides). Hence, we have
. Now
. Hence, we have
. This is true for all
. Hence, all the
, which proves that we have a direct sum decomposition of
.
Why is all this relevant? Because using the minimal polynomial of any transformation
, we can construct such
‘s which satisfy the above two conditions, and can hence decompose the vector space as a direct sum of
subspaces. Moreover, these subspaces have the additional property that they’re
-invariant. Each