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 ...

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.

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 ...

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 ...

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 (...

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 ...

6
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 ...

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).
...

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 $\max(b_1, b_n) - \lambda_\max$.

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)^{-...

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-...

4
votes

Accepted

### Eigenvalues of a specific matrix

For the signed circulant matrix
$$U:=\left[\begin{matrix}
& 1 & & & \\
& & \ddots & & \\
& & & 1 &\\
-1 & & & &
\end{matrix}\...

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=...

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 ...

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.

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 ...

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 ...

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, ...

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>...

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 ...

2
votes

### Non-singular matrix with restricted entries

This is perhaps a minor observation (Edit: thanks to Peter Taylorâ€™s comment below).
If $S$ has every admissible symmetric matrix being non-singular, then $-1\notin S$.
This is due to the singular $2\...

2
votes

### Eigenvalues and eigenvectors of k-blocks matrix

$X$ is simply a tensor product $C\otimes D$ where $C$ is the matrix with all diagonal entries $a$ and non-diagonal entries $b$ and where each entry in $D$ is $1$.
If $R$, $S$ are diagonalizable, then ...

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, ...

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 ...

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 ...

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 $i$.
...

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 $ALA$ ...

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