25
votes
Accepted
Proof that block matrix has determinant $1$
Write the SVD of $A$, say $A=PDQ^T$ with $D$ diagonal with non-negative entries and $P\in O(n),Q\in O(m)$. Then $\sqrt{I_n + AA^T} = P\sqrt{1+D^2}P^T$
and $\sqrt{I_m+ A^TA} = Q\sqrt{1+D^2}Q^T$. This ...
- 1,724
22
votes
Proof that block matrix has determinant $1$
We have $Af(A^TA)=f(AA^T)A$ for any reasonable function $f$, including $f(x)=\sqrt{1+x}$. This suffices to check for $f(x)=x^k$ when it is obvious, then approximate your function by a polynomial.
- 109k
15
votes
Accepted
When does the determinant distribute over addition?
let me assume $A$ is invertible, then you ask when
$$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$
so if $X$ has eigenvalues $x_i$, $i=1,2,\ldots n$, you would need
$$\prod_{i}(1+x_i)=1+\prod_i x_i$$
basically ...
- 188k
14
votes
Accepted
Off-diagonalize a matrix
This is a so-called chiral symmetry. The restriction on the symmetry of the spectrum of $M$ is the only restriction you need, you can then bring $M$ to the desired off-diagonal form by a unitary ...
- 188k
11
votes
Conjugated subgroups in $\mathsf{GL}(m+n,\mathbb{Z})$ implies conjugated subgroups in $\mathsf{GL}(n,\mathbb{Z})$?
$\def\ZZ{\mathbb{Z}}\def\GL{\text{GL}}$We can make partial progress using:
Warfield, R. B. jun., Cancellation of modules and groups and stable range of endomorphism rings, Pac. J. Math. 91, 457-485 (...
- 156k
9
votes
Accepted
One question on block-circulant matrices
The formula for the specific case is
$$\det K=\det(A+B+C+D)\det(A-B+C-D)\det(A+iB-C-iD)\det(A-iB-C+iD).$$
More generally, for a block-circulant matrix with $n$ square blocks $A_0,\ldots,A_{n-1}$, the ...
- 52.3k
6
votes
Accepted
Block matrices and their determinants
Flip the order of the Kronecker products to get $M'=A_n(I_n,I_n)+B_n\otimes T_n$, where $T_n=A_n(1,0)$. Note that $\det M=\det M'$. Since all blocks are polynomial in $A$, they commute, and therefore ...
- 1,593
4
votes
Accepted
Spectrum of this block matrix
If $\lambda_\max$ is the greatest eigenvalue of $T$, the least eigenvalue of $A$ is between $-\lambda_\max$ and $\max(b_1, b_n) - \lambda_\max$.
- 54.2k
4
votes
Accepted
Is there a formula for the determinant of a block matrix of this kind?
Just a sketch of an idea that seems to work:
You can get rid of the corrections $C$ using the matrix determinant lemma (or, better, replace them with $B$, which makes the matrix block circulant).
...
- 20.2k
4
votes
Accepted
A closed-form expression for the inverse of a block-matrix
Say that
$$B^{-1}=:\begin{pmatrix} b & X^T \\ Y & M \end{pmatrix}.$$
Then using Schur's complement formula (thanks to Nathaniel), $b=(x-{\bf1}^TA^{-1}{\bf1})^{-1}$ and $M=(A-x^{-1}{\bf11}^T)^{-...
- 52.3k
4
votes
Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks
To see what you might expect for a relation, consider the case of a $2\times 2$ matrix $M=\begin{pmatrix}a&b\\
c&d\end{pmatrix}$, with eigenvalues $\lambda_\pm=\tfrac{1}{2}(a+d)\pm\sqrt{4bc+(a-...
- 188k
4
votes
Accepted
Eigenvalues of a specific matrix
For the signed circulant matrix
$$U:=\left[\begin{matrix}
& 1 & & & \\
& & \ddots & & \\
& & & 1 &\\
-1 & & & &
\end{matrix}\...
- 10.1k
4
votes
Accepted
Solving a recursion for polynomials defined by a matrix product
Your polynomial is precisely
$$
\sum_{k_1+2k_2+\cdots+nk_n=n}\binom{k_1+\cdots+k_n}{k_1,\ldots,k_n}X_1^{k_1}\cdots X_n^{k_n}.
$$
The proof is straightforward by induction: you have
$$
p_n(X)=\sum_{i=...
- 16.9k
3
votes
Accepted
The normalizer of block diagonal matrices
By request, from my comment: Your guess is correct. Because there are clearly elements in the normaliser arbitrarily permuting same-sized blocks, if you've got an element of the normaliser, you may ...
- 12.9k
3
votes
When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?
$$B=\left[\matrix{1&1&1&1\\1&1&-1&-1\\-1&-1&-1&1\\-1&1&-1&1}\right]$$
and probably many other solutions. I'm also voting to close because you didn't ...
- 37.7k
3
votes
Accepted
Conditions to solve linear system with matrix blocks
You find various conditions in Section 3 of the classical review paper by Benzi-Golub-Liesen on this kind of problems, which are known as saddle-point problems.
- 20.2k
3
votes
Accepted
Tight upper bound on a ratio involving symmetric PSD matrices and Kronecker products
There is no upper for $m$ that is independent of $d$ and $T$. In fact, for a fixed $d$, the best upper bound that works for all $T$ is of order $\sqrt{d}$, and this is already tight for $T=d$.
The ...
- 9,486
2
votes
Non-singular matrix with restricted entries
[EXPANDED]
PART 1 (also done by Peter)
If $x,y$ are coprime and have opposite sign, there is a singular symmetric matrix with 1 on the diagonal and
only $x$ and $y$ off the diagonal.
Say $x<0,y>...
- 37.7k
2
votes
Non-singular matrix with restricted entries
Disclaimer: this is only a partial answer.
If $S = \{x, y\}$ (considered as variables), the determinant must be a polynomial with integer coefficients and constant coefficient 1. Therefore by Gauss's ...
- 7,226
2
votes
Non-singular matrix with restricted entries
This is perhaps a minor observation (Edit: thanks to Peter Taylor’s comment below).
If $S$ has every admissible symmetric matrix being non-singular, then $-1\notin S$.
This is due to the singular $2\...
- 1,453
2
votes
Accepted
Inverse of a larger matrix where the inverse of the submatrix is known
You know $\begin{pmatrix} A & 0 \\ 0 & 1 \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} & 0 \\ 0 & 1 \end{pmatrix}$, and from there you can make two successive rank-$1$ modifications, ...
- 6,237
2
votes
Accepted
Solve linear system with bordered positive definite matrix
By combining the useful comments of Rodrigo and Todd, the methodology to solve this system is shown here below. One caveat is that the method is probably not very efficient, since you need to use the ...
- 139
1
vote
Eigenvalues in unit disk for a 2×2 block matrix
Clearly all eigenvalues apart from the eigenvalue $\lambda(\epsilon)$ with $\lambda(0) = 1$ stay in the open unit disk for $\epsilon$ sufficiently small. To see what happens to the last eigenvalue, ...
- 20.2k
1
vote
Eigenvalues in unit disk for a 2×2 block matrix
Part 1 - Determinant of $X$
As a partial result, it is possible to show that $\lvert\det X\rvert < 1$ for sufficiently small $\epsilon$. In fact, because $B$ and $C$ commute, then thanks to a known ...
- 205
1
vote
Accepted
Eigenvalues of a block matrix with zero diagonal blocks
If you decompose $M=\begin{pmatrix} X_{q\times q}&Y_{q\times k_3}\\ (Y_{q\times k_3})^{\rm T}&0_{k_3\times k_3}\end{pmatrix}$ into four block matrices, with $q=k_1+k_2$, then the determinant ...
- 188k
1
vote
Sufficient conditions for invertibility of a block tridiagonal matrix
The following is a list of answers I know for some specific cases. However, they are not strong enough for my uses.
Simple conditions
A sufficient, but weak condition is that $C_i = 0$ for each $i$.
...
- 397
1
vote
Accepted
If the direct sum of $L$ and $M$ has a pseudoinverse, then do $L$ and $M$ have pseudoinverses?
I propose an answer to my own question:
The following observation is based on Section 3.1 of "Theory of generalized inverses over commutative rings" by K.P.S. Bhaskara Rao.
The matrix $ALA$ ...
- 4,943
1
vote
Determinant diagonal blocks compound matrix
There is a general theorem, which can be proved by using Schur's complement formula: let a matrix $A\in M_{pq}(k)$ be written blockwise, with blocks $A_{ij}\in M_q(k)$ for $1\le i,j\le p$. Assume that ...
- 52.3k
Only top scored, non community-wiki answers of a minimum length are eligible