All Questions
129 questions
3
votes
0
answers
39
views
A non-singularity property for sets of real matrices
Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
- 943
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 82.6k
5
votes
0
answers
91
views
elementary matrices over a regular ring
Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
- 3,472
6
votes
0
answers
259
views
Diameter of finite rational matrix groups
Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$.
For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
- 63.9k
3
votes
0
answers
231
views
Singularity of symmetric block matrix with singular diagonal blocks
One can show that the following statement holds:
Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank,
$$\begin{bmatrix}A & B \\ B^T &...
CommunityBot
- 1
7
votes
0
answers
658
views
Row rank and column rank of matrix with entries in a commutative ring
Let $R$ be a unital commutative ring and $A\in M_{n\times m}(R)$. Under which appropriate invariant "rank" of modules discussed
in "Ranks of Modules"
one can say that the row rank of $A$ is ...
- 63.9k
3
votes
1
answer
454
views
Difference of pseudoinverse bound under assumptions
This seems like a standard problem, but unable to find a solution online.
Suppose we have two singular PSD matrices A and B with the following assumptions:
$ 0 < x \leq ||A|| \leq y$
$ 0 < ||...
- 228
12
votes
2
answers
2k
views
Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...
- 51
3
votes
0
answers
180
views
Automorphisms of infinite matrix algebra
This is a similar question to one that I posted in MSE a few days ago.
I recently came across this paper from Alahmedi, Alsulami, Jain and Zelmanov, which quoted the following result for $M_\infty(K)$...
- 63.9k
7
votes
1
answer
2k
views
Determinant and Inverse of a Toeplitz matrix
Let $T(n,k)$ be a $n \times n$ symmetric Toeplitz matrix, where all the entries of first $k$ super-diagonal (and sub-diagonal), last $k-1$ super-diagonal (and sub-diagonal) are ones, and rest of the ...
CommunityBot
- 1
10
votes
0
answers
237
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 4,729
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 66.3k
11
votes
1
answer
896
views
Decide if a matrix is transposable
A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...
- 4,729
0
votes
1
answer
243
views
A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]
For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
CommunityBot
- 1
3
votes
0
answers
117
views
Sparsest similar matrix
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...
- 4,729
3
votes
3
answers
504
views
Reference request. Something like a rank for matrices over non-commutative rings
I'm looking for some function $P: M \rightarrow R$, where $M$ is matrix ring over some non-commutative ring that would behave "like" rank. In particular:
$P(A+B) \leq P(A)+P(B)$
$P(A\times B) \leq P(...
- 21
1
vote
1
answer
1k
views
Expected size of determinant of $AA^T$ for non-square random $A$
If $A$ is chosen uniformly at random over all possible $m \times n$ (0,1)-matrices, what is the expected size of the absolute value of the determinant of $AA^T$. We can assume $m < n$ and all ...
- 4,729
0
votes
1
answer
3k
views
Cholesky decomposition – non-positive definite matrix
In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
- 4,729
3
votes
1
answer
1k
views
The largest eigenvalue of a binary matrix with specific density
I would like to find the largest eigenvalue of an $n \times n$ binary matrix of density $p$, i.e., with $p n^{2}$ ones and $(1-p) n^{2}$ zeros. Any idea or reference is welcome.
- 4,729
1
vote
0
answers
26
views
Depth bound of generating an matrix algebra
Let $X$ be Hilbert spaces $\mathbb{C}^d$, and $L(X)$ be the sets of linear operators of $X$. We are given a matrix subspace $S\subset L(X)$. Via the following procedure, one can generate the smallest ...
- 1,503
6
votes
0
answers
223
views
Specifying cokernels of all powers of $p$-adic matrix
Given a matrix $A \in M_d(\mathbb{Z}_p)$ (with nonzero determinant), viewed as a map $\mathbb{Z}_p^d \to \mathbb{Z}_p^d$, I am interested in the sequence of abelian $p$-groups $\{coker(A^n)\}_{n \geq ...
- 63.9k
2
votes
1
answer
389
views
On the notion of Hadamard rank of matrix
Given a matrix $M\in\Bbb F_2^{n\times n}$, define its Hadamard rank $h(M)$ to be the minimum number of rank $\leq2$ matrices in $\Bbb F_2^{n\times n}$ with Hadamard product (that is, the entry-wise ...
- 63.9k
9
votes
1
answer
804
views
A singular value-eigenvalue inequality
Singular value or eigenvalue problems lie at the center of matrix analysis. One classical result is
$$\lambda_{j}(X^{*}X+Y^{*}Y)\geq 2\sigma_j(XY^*)$$
for $j \in \{1, \ldots, n\}$, where $\lambda_j(\...
- 2,753
3
votes
0
answers
99
views
Are there nondiagonal ring homomorphisms $f:R\rightarrow M_n(R)$ for an integral domain R?
Let $R$ be an integral domain and $M_n(R)$ the ring of $n\times n$ matrices over $R$.
Question 1: Is there a ring homomorphism $f:R\rightarrow M_n(R)$ such that $f(r)$ is a nondiagonal matrix for ...
- 139
2
votes
1
answer
245
views
Are symmetric matrices a maximal subspace of conjugate-normal matrices
Let $M$ denote the space of all complex $n\times n$ matrices. ($M$ is a complex vector space.)
Let $C\subseteq M$ be the set of all conjugate-normal matrices (i.e., matrices $A$ with $AA^*=\overline{...
- 3,984
2
votes
1
answer
799
views
Relation between LDLT and eigendecomposition of real symmetric matrices
The wikipedia page for Cholesky decomposition says:
For real matrices, the factorization has the form $A = LDL^T$ and is
often referred to as LDLT decomposition. It is
closely related to the ...
- 19.6k
41
votes
4
answers
33k
views
What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 415
17
votes
0
answers
704
views
When is the determinant an $8$-th power?
I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
- 7,300
5
votes
0
answers
315
views
Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.9k
1
vote
0
answers
95
views
Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?
Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring.
I want to prove
that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
3
votes
1
answer
236
views
For $T$ the $2\times 2$ triangular matrices over $R$, can we write $GL_2(T)=U(T)E_2(T)$?
Let $R$ be a commutative ring with identity, and let $T = T_2(R)$ be the ring of $2\times 2$ upper triangular matrices over $R$. Is it true that the following identity holds?
$$GL_2(T)=U(T)E_2(T)$$
...
- 7,893
2
votes
2
answers
117
views
Powers of small square matrices over the Laurent polynomial ring with integer coefficients
I'm trying to calculate the powers of a 2 by 2 matrix with entries in $\mathbb{Z} \left[ t,t^{-1} \right]$.
The matrix is \begin{bmatrix}
0 & 1 \\
1 & t
\end{bmatrix}
I tought of writing my ...
- 43.2k
51
votes
1
answer
2k
views
Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...
- 12.9k
2
votes
2
answers
242
views
Integer square $2 \times 2$ block matrix inverse
Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix
$$
\mathbf{M} =
\left(
\begin{array}{cc}
\mathbf{A} & \mathbf{B} \\
\mathbf{C} & \mathbf{D}
\end{array}
\right) ,
$$
where $\...
3
votes
1
answer
101
views
Isomorphism concerning $Soc(M_n(R))$
It is known that $M_n(R/J(R))\simeq M_n(R)/M_n(J(R))=M_n(R)/J(M_n(R))$. I tried to prove the same "isomorphism" replacing $J(R)$ by $Soc(R_R)$, where $J(R)$ and $Soc(R_R)$ stand for the Jacobson ...
- 5,039
4
votes
0
answers
112
views
Inducing surjections on $GL_n(-)$?
Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $...
- 63.9k
7
votes
1
answer
212
views
What does $K_1(R)$ tell us about $GL_n(R)/E_n(R)$?
Let $D$ be a division ring, and $R=D[t_1,\ldots,t_n]$. If $GL_m(R)$ is the usual group of invertible matrices over $R$, then by $E_m(R)$ I mean the subgroup of $GL_m(R)$ generated by the elementary ...
- 3,186
7
votes
1
answer
517
views
When is $GL_m(R)$ generated by elementary and diagonal matrices?
Let $D$ be a division ring and $R=D[t_1,\ldots,t_n]$ the polynomial ring in $n$ variables. Now let $GL_m(R),\,E_m(R)$ be the usual general linear group and its subgroup generated by the elementary ...
- 348
2
votes
1
answer
83
views
Exist matrices such that $f(X) = \sum_{j = j}A_jXB_j$ for all $X \in \text{M}_N(\mathbb{C})$? [closed]
Let $f: \text{M}_n(\mathbb{C}) \to \text{M}_n(\mathbb{C})$ be a $\mathbb{C}$-linear map (not necessarily an algebra homomorphism). Do there exist matrices $A_1, \ldots, A_d \in \text{M}_n(\mathbb{C})$ ...
- 1,996
4
votes
1
answer
161
views
Subring of matrix algebra with common eigenvectors
Consider the matrix algebra $\mathcal{M}_n(\mathbb{C})$ (acting on $n$ dimensional space $V$) and let $R$ be subring of matrices of $\mathcal{M}_n(\mathbb{C})$.
Suppose that any two elements of $R$...
- 63.9k
10
votes
1
answer
3k
views
Reverse Minkowski (and related) Determinant Inequalities
For positive semidefinite matrices $A,B,C \in \mathbb{R}^{n\times n}$, the following inequalities are well known:
$$(\det(A+B))^{1/n} \geq (\det A)^{1/n} + (\det B)^{1/n} $$
and
$$\det(A+B+C) + \...
CommunityBot
- 1
22
votes
1
answer
33k
views
vector to diagonal matrix [closed]
For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector.
Is there a simple way to write this transformation ...
- 19.6k
0
votes
2
answers
149
views
Simultaneous special orthogonal similarity problem
Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
$$A=...
CommunityBot
- 1
7
votes
1
answer
1k
views
The saturation of Murray von Neumann relation
Edit: According to comment of Pace Nielsen, I remove question 2 of the previous version:
Let $R$ be a unital ring. We define Murray Von Neumann relation $M$ on $R$ as follows:
We say $a M b$ iff $...
CommunityBot
- 1
3
votes
3
answers
357
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
- 19.6k
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
CommunityBot
- 1
-4
votes
2
answers
357
views
Representing quaternions as matrices [closed]
Assume F is a field of characteristic different than 2. Let a, b be invertible elements in F, and let A(a,b) be the generalised quaternions. Using the Artin–Wedderburn theorem, there is a ...
- 23k
5
votes
2
answers
338
views
Polynomial identities for mod p matrices
Can there be a polynomial over the field $F_p$ of $p$ elements ($p$ prime) in non-commuting variables $X_1,..., X_r$ such that:
1) $f(A_1,...,A_r)=0$ for every $n \times n$ matrices $A_1,...,A_r$ ...
- 398
6
votes
2
answers
4k
views
Characteristic polynomial of Kronecker/tensor product
This was asked before on stackexchange but no answer was given. The question is the following:
Let $A$ and $B$ be matrices in $GL(n)$ and $GL(m)$ respectively. Their tensor product $A\otimes B$ is ...
CommunityBot
- 1
10
votes
2
answers
1k
views
When the determinant of a 2x2 polynomial matrix is a square?
Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
CommunityBot
- 1