Eigenvalues of block matrices with upper triangular blocks
Jan 31, 2021
This past week, Rujun Jiang and I posted a
preprint on almost simultaneously
diagonalizable quadratic forms (and other related ideas). At some point
in the future, I will try write a short expository blog post about the
main contributions of this work. Today, I’m simply going to present a
factI don’t claim that this is new in any way and imagine
this fact is known somewhere. On the other hand, I was personally unable
to find a reference for it and would appreciate if anyone had pointers
for me! about the eigenvalues of block matrices with upper
triangular blocks that I needed/learned/proved while working on this
First, let’s define the sets of matrices we care about.
Definition 1. A matrix T∈Cn×m is upper
triangular if m−j+i>min(n,m)⟹Ti,j=0.
Pictorially, an upper triangular matrix looks like ⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛∗0⋮0⋯⋱⋯⋱⋯∗⋮∗0⋮0⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞or⎝⎜⎜⎛0⋮0⋯⋱⋯0⋮0∗⋯⋱∗⋮∗⎠⎟⎟⎞ when n≥m and n≤m respectively.
Suppose (n1,…,nk) is such that ∑pnp=n. Given
T∈Cn×n, we will let [T]p,q denote the
(p,q)th block of T so that
Proposition 1. Let (n1,…,nk) such that ∑pnp=n and
T∈Cn×n such that [T]p,q is upper triangular
for all p,q∈[k]. Then, the determinant of T depends only on the
diagonal entries of blocks [T]p,q where np=nq.
Pictorially (for the case (n1,n2,n3)=(2,2,3)), this proposition
says that the determinant of a matrix with support ⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛@@∗∗@∗@∗∗@@∗∗@∗@∗∗@∗∗∗@∗∗∗∗∗∗@⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞
depends only on the entries in the @ positions.
At a high level, this proposition states that the determinant (whence
also the characteristic polynomial and eigenvalues) of a block matrix
with upper triangular blocks depends on only a very small number of
entries of the block matrix.
I will now present a really neat proof of this fact using a proof
strategy that Kevin Pratt suggested.
Proof. We will show the
strongerThis stronger statement in fact implies that the
permanent, or in fact any generalized matrix function, of such a
matrix depends only on a small number of entries. statement: For
any permutation σ∈Sn, if ∏iTi,σ(i) is
nonzero, then for every i, we have that (i,σ(i)) is a diagonal
entry of some block [T]p,q where np=nq.
First assign a weight wi to each i∈[n]: Given i∈[n], let p
denote the block containing i and let r denote position of i
within the pth block. Set wi:=r−np/2. Next for each i,j∈[n],
assign the weight wi,j:=wj−wi.
Equivalently, if i,j corresponds to p,q,r,s, then set
Next, as T has upper triangular blocks, we also have that
We deduce that if Ti,j=0, then wi,j≥0. Furthermore, if
Ti,j=0 and wi,j=0, then it must be the case that
np=nq and s=r.
Finally, note that for any permutation σ∈Sn, we have
This concludes the proof as then if ∏iTi,σ(i) is
nonzero, it must be the case that for all i∈[n], we have
Ti,σ(i)=0 and wi,σ(i)=0. ◻
Intuitively, this proof looks at a matrix with upper triangular blocks
and compares its support with some carefully constructed weighting of
its entries. For (n1,n2,n3)=(2,2,3), this looks like
⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞,⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛0−10−10.5−0.5−1.510101.50.5−0.50−10−10.5−0.5−1.510101.51.5−0.5−0.5−1.5−0.5−1.50−1−20.5−0.50.5−0.510−188.8.131.52.5210⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞. The proof then observes that: The
support of T corresponds exactly to the entries with nonnegative
weight. Additionally, the intersection of the support of T with the
entries of weight zero is exactly the set of diagonal entries of blocks
[T]p,q where np=nq.
In the setting of our paper, we have that the blocks of T are
furthermore Toeplitz, i.e., their entries are constant along each
diagonal. Then using the above proposition (along with other more
elementary facts), we are then able to make crazy-looking statements
such as ⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛ac∗∗a∗c∗∗bd∗∗b∗d∗∗e∗∗∗e∗∗∗∗∗∗e⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞=σ⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛acacbdbdeee⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎞=σ⎝⎜⎛acbde⎠⎟⎞, where =σ indicates
that the matrices have the same spectrum.