ST and TS have the same eigenvalue

[maNoFchangE]

Homework Statement

Prove that, if $T,S\in \mathcal{L}(V)$ then $TS$ and $ST$ have the same eigenvalues.

Homework Equations

The Attempt at a Solution

Suppose $T$ is written in a basis in which its matrix is upper triangular, and so is $S$ (these bases may be of different list of vectors in $V$). Since both $T$ and $S$ are upper triangular, $TS$ and $ST$ are also upper triangular. Now, the diagonal element of $TS$ is
$$
\sum_i T_{pi}S_{ip}
$$
which is the same as the diagonal element of $ST$
$$
\sum_i S_{pi}T_{ip}
$$
Therefore, $ST$ and $TS$ have the same eigenvalues.
What bothers me is my starting assumption; taking $T$ and $S$ in their own upper triangular bases. Since these bases can be a different list of vectors, is it alright then to multiply their matrices?

 

[Samy_A]

Homework Statement

Prove that, if $T,S\in \mathcal{L}(V)$ then $TS$ and $ST$ have the same eigenvalues.

Homework Equations

The Attempt at a Solution

Suppose $T$ is written in a basis in which its matrix is upper triangular, and so is $S$ (these bases may be of different list of vectors in $V$). Since both $T$ and $S$ are upper triangular, $TS$ and $ST$ are also upper triangular. Now, the diagonal element of $TS$ is
$$
\sum_i T_{pi}S_{ip}
$$
which is the same as the diagonal element of $ST$
$$
\sum_i S_{pi}T_{ip}
$$
Therefore, $ST$ and $TS$ have the same eigenvalues.
What bothers me is my starting assumption; taking $T$ and $S$ in their own upper triangular bases. Since these bases can be a different list of vectors, is it alright then to multiply their matrices?

Assume $\lambda$ is an eigenvalue of $TS$.
Then, by definition of an eigenvalue, there exists an $x\in V,\ x\neq \vec0$ such that $TSx=\lambda x$.
Now try to prove that $\lambda$ is an eigenvalue of $ST$.

(You don't need a matrix representation of the operators.)

 

[maNoFchangE]

Multiplying
$$
TSx=\lambda x
$$
with $S$, I get
$$
ST(Sx) = \lambda (Sx)
$$
Alright, that looks good. But how do I prove that $Sx$ is not zero given $x \neq 0$?

 

[Samy_A]

Multiplying
$$
TSx=\lambda x
$$
with $T$, I get
$$
ST(Tx) = \lambda (Tx)
$$
Alright, that looks good. But how do I prove that $Tx$ is not zero given $x \neq 0$?

You can't. There is no reason why $Tx$ couldn't be the 0 vector.
Hint: multiply from the other side.

 

[maNoFchangE]

I have edited my post.

You can't.

Why can't I have $Sx=0$? What if $x\in \textrm{null }S$?

 

[Samy_A]

Multiplying
$$
TSx=\lambda x
$$
with $S$, I get
$$
ST(Sx) = \lambda (Sx)
$$
Alright, that looks good. But how do I prove that $Sx$ is not zero given $x \neq 0$?

Assume $Sx=\vec 0$. What does that imply, given that $TSx=\lambda x$?

 

[maNoFchangE]

Assume $Sx=\vec 0$. What does that imply, given that $TSx=\lambda x$?

That means $\lambda x=0$, since $x \neq 0$ then $\lambda=0$. But what does it say about $Sx$ being not 0?

 

[Samy_A]

That means $\lambda x=0$, since $x \neq 0$ then $\lambda=0$. But what does it say about $Sx$ being not 0?

Take the case $\lambda \neq 0$ first. What have you then proven?

 

[maNoFchangE]

Take the case $\lambda \neq 0$ first. What have you then proven?

In that case $x=0$, which disproves the starting hypothesis that $x\neq 0$. Therefore in this case, $Sx$ cannot be zero.
But in the case $\lambda=0$, what next?
It does not seem to violate any hypothesis we have assumed.

 

[Samy_A]

In that case $x=0$, which disproves the starting hypothesis that $x\neq 0$. Therefore in this case, $Sx=0$.
But in the case $\lambda=0$, what next?
It does not seem to violate any hypothesis we have assumed.

Ok, so you have proven that $TS$ and $ST$ have the same non-zero eigenvalues.

Now for the tricky case $\lambda =0$.
Still assuming $x \neq \vec 0$, you have $TSx=\vec0$, and thus $ST(Sx)=\vec 0$.
If $Sx \neq \vec 0$, then we are done.

If $Sx =\vec 0$, $S$ is not injective.
At this point, you need to take into account that $V$ is finite dimensional (you didn't specifically state this, but since you used matrices, I assume that it is the case).
If $S$ is not injective, it is not surjective. What does that tell you about $ST$?

 

[maNoFchangE]

If $S$ is not surjective, then $ST$ is not surjective either.
Sorry I just can't see where this is leading to. If $ST$ is not surjective, is it inconsistent with one of our assumptions?

 

[Samy_A]

If $S$ is not surjective, then $ST$ is not surjective either.
Sorry I just can't see where this is leading to. If $ST$ is not surjective, is it inconsistent with one of our assumptions?

Correct, $ST$ is not surjective.
Can it be injective? If not, what does that imply as far as eigenvalues of $ST$ are concerned?

 

[maNoFchangE]

Correct, $ST$ is not surjective.
Can it be injective? If not, what does that imply as far as eigenvalues of $ST$ are concerned?

Since $ST$ is an operator in a complex vector space, if it's not surjective, it will not be injective nor it be invertible, am I right?
Since $ST$ is not invertible, there must be at least one of its eigenvalues equal to $0$. But isn't it what we were starting from? In post #10, we have set $\lambda=0$, aren't we just circulating here?

 

[Samy_A]

Since $ST$ is an operator in a complex vector space, if it's not surjective, it will not be injective nor it be invertible, am I right?
Since $ST$ is not invertible, there must be at least one of its eigenvalues equal to $0$. But isn't it what we were starting from? In post #10, we have set $\lambda=0$, aren't we just circulating here?

No, in post #10 we started with an eigenvalue of $TS$. You now have proved that if 0 is an eigenvalue of $TS$, it also is an eigenvalue of $ST$.
We are not circulating, we went from $TS$ to $ST$.

Notice that for the case of eigenvalue 0, $V$ has to have finite dimension. If $V$ is an infinite dimensional vector space, it is possible for $TS$ to have 0 as eigenvalue, while $ST$ is injective.

 

[maNoFchangE]

I see! I was not aware that we have been switching to calculating the eigenvalue of $ST$. So, $\lambda=0$ is also an eigevalue of $ST$, and the corresponding eigenvector is zero, is that right?

 

[Samy_A]

I see! I was not aware that we have been switching to calculating the eigenvalue of $ST$.

Yes, that is the whole purpose of this exercise.

So, $\lambda=0$ is also an eigevalue of $ST$, and the corresponding eigenvector is zero, is that right?

It is an eigenvalue, but we can't say anything about the eigenvector.
Certainly the eigenvector is not the zero vector. An eigenvector is, by definition, never the zero vector.
For a simple reason: $T\vec 0=\lambda \vec 0$ is true for any linear operator $T$ and any scalar $\lambda$.

 

[maNoFchangE]

Ok many thanks for the help.
Just one more thing, my original question was actually is it allowed to multiply two square matrices which are represented in different bases?

 

[Samy_A]

Ok many thanks for the help.
Just one more thing, my original question was actually is it allowed to multiply two square matrices which are represented in different bases?

No, you can't just do that. You can of course multiply the matrices, but the resulting matrix is not necessarily the matrix representing the composition of the two operators (for what basis should that be?)

 

[maNoFchangE]

No, you can't just do that. You can of course multiply the matrices, but the resulting matrix is not necessarily the matrix representing the composition of the two operators (for what basis should that be?)

Ah, that makes sense.