Dtjytyk

Let $A$ be a block upper triangular matrix:

$$A = begin{pmatrix} A_{1,1}&A_{1,2}\ 0&A_{2,2} end{pmatrix}$$

where $A_{1,1} ∈ C^{p times p}$, $A_{2,2} ∈ C^{(n-p) times (n-p)}$. Show that the eigenvalues of $A$ are the combined eigenvalues of $A_{1,1}$ and $A_{2,2}$

I've been pretty much stuck looking at this for a good hour and a half, so any help would be much appreciated. Thanks.

Let $A$ be the original matrix of size $n times n$. One way out is to use the identity. (Result from Schur Complement) https://en.wikipedia.org/wiki/Schur_complement

$det left( begin{matrix} B_{1,1}&B_{1,2}\ B_{2,1 }&B_{2,2} end{matrix} right) = det(B_{1,1}) times det(B_{22} - B_{21}B_{11}^{-1}B_{12})$.

We know that $lambda$ is a number such that $Ax = lambda x$. From which we get $det(A-lambda I_n) = 0$.

In your case, the matrix $A_{21}$ is a zero matrix and hence, we get $det(A-lambda I_n) = det left( left( begin{matrix} A_{1,1}&A_{1,2}\ 0&A_{2,2} end{matrix}right) - lambda I_n right) = det left( begin{matrix} A_{1,1} - lambda I_{k}&A_{1,2}\ 0&A_{2,2} - lambda I_{n-k} end{matrix}right)$

Hence $det(A-lambda I_n) = det(A_{1,1} - lambda I_{k}) times det(A_{22} - lambda I_{n-k})$.

So we get that if $lambda$ is an eigen value of $A_{11}$ or $A_{22}$, then either $det(A_{11}-lambda I_k) = 0$ or $det(A_{22}-lambda I_{n-k}) = 0$ and hence $det(A-lambda I_n) = 0$ and hence $lambda$ is an eigenvalue of $A$.

Similarly, if $lambda$ is an eigenvalue of $A$, then $det(A-lambda I_n) = 0$, then either $det(A_{11}-lambda I_k) = 0$ or $det(A_{22}-lambda I_{n-k}) = 0$ and hence $lambda$ is an eigen value of $A_{11}$ or $A_{22}$.

Edit

There is actually a small error in the above argument.

You might wonder that if $lambda$ is an eigenvalue of $A_{11}$, then $A_{11} - lambda I_k$ is not invertible and hence the identity $det left( begin{matrix} B_{1,1}&B_{1,2}\ B_{2,1 }&B_{2,2} end{matrix} right) = det(B_{1,1}) times det(B_{22} - B_{21}B_{11}^{-1}B_{12})$ is false since $B_{11}$ is not invertible.

However, there is an another identity $det left( begin{matrix} B_{1,1}&B_{1,2}\ 0&B_{2,2} end{matrix} right) = det(B_{1,1}) times det(B_{22})$ which is always true. (Prove both the identites as an exercise).

We can make use of this identity to get
$det(A-lambda I_n) = det(A_{1,1} - lambda I_{k}) times det(A_{22} - lambda I_{n-k})$.

A simpler way is from the definition. Is is easy to show that if $lambda_1$ is an eigenvalue of the upper diagonal block $A_{1,1}$, with eigenvector $p_1$, (size $n_1$) then it's also an eigenvalue of the full matrix, with the same eigenvector augmented with zeros.

$A_{1,1} ; p_1 = lambda_1 p_1$ with $p_1 ne 0 $

So

$$ left( begin{matrix} A_{1,1}&A_{1,2} \ 0 &A_{2,2} end{matrix} right)
left( begin{matrix} p_1 \ 0 end{matrix} right) =
left( begin{matrix} A_{1,1} ; p_1 \ 0 end{matrix} right) =
left( begin{matrix} lambda_1 p_1 \ 0 end{matrix} right) =
lambda_1 left( begin{matrix} p_1 \ 0 end{matrix} right) $$

Hence if $lambda$ is eigenvalue of $A_{1,1}$ then it's also eigenvalue of $A$. There are $n_1$ (counting multiplicity) such eigenvalues. The same applies to the lower diagonal block $A_{2,2}$. So we have found the $n_1$ + $n_2 = n$ eigenvalues of the full matrix. (Wrong! This only applied to block diagonal matrix - Fixed below)

Suposse now that $lambda_2$ is eigenvalue of $A_{2,2}$ with eigenvector $p_2$.

If $lambda_2$ is also eigenvalue of $A_{1,1}$, we have proved above that it's also eigenvalue of $A$. So, let's assume it's not eigenvalue of $A_{1,1}$ - hence $|A_{1,1} - lambda_2 I|ne 0$. Now

$$left( begin{matrix} A_{1,1}&A_{1,2} \ 0 &A_{2,2} end{matrix} right)
left( begin{matrix} x \ p_2 end{matrix} right) =
left( begin{matrix} A_{1,1} x + A_{1,2} p_2 \ lambda_2 p_2 end{matrix} right)
$$
We can make $ A_{1,1} x + A_{1,2} p_2 = lambda_2 x$ by choosing $x = - (A_{1,1} - lambda_2 I)^{-1} A_{1,2} ; p_2$; and so we found an eigenvector for $A$ with $lambda_2$ as eigenvalue.

It this way, we showed that if $lambda$ is eigenvalue of $A_{1,1}$ or $A_{2,2}$, then it's an eigenvalue of $A$.

To complete the proof, one should show the other way round: that if $lambda$ is eigenvalue of $A$ then it's eigenvalue of $A_{1,1}$ or $A_{2,2}$. But that's easy:

$$left( begin{matrix} A_{1,1}&A_{1,2} \ 0 &A_{2,2} end{matrix} right)
left( begin{matrix} x_1 \ x_2 end{matrix} right) =
left( begin{matrix} A_{1,1} ; x_1 + A_{1,2} ; x_2 \ A_{2,2} ; x_2 end{matrix} right)
= left( begin{matrix} lambda ; x_1 \ lambda ; x_2 end{matrix} right)
$$

Now, either $x_2 = 0$ or not. If not, then $lambda$ is eigenvalue of $A_{2,2}$. If yes,
it's eigenvalue of $A_{1,1}$.

For another approach for a proof you can use the Gershgorin disc theorem (sometimes Hirschhorn due to pronounciation differences between alphabets) to prove the disks for the individual matrices are the same as the discs for the large matrix so the sets of possible eigenvalues must be the same. This is because the radial contribution to the disks are 0 all over all entries for the lower left block since $|0| = 0$ and $0+0=0$.