All Questions
Tagged with determinants matrices
42 questions with no upvoted or accepted answers
16
votes
0
answers
809
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
12
votes
0
answers
508
views
More mysterious properties of Gram matrix
This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question.
The following fact could be extracted from 0402087:
For any $a_i\...
- 4,316
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 $...
- 909
8
votes
0
answers
176
views
Nonzero subdeterminants conjecture: has anybody seen this anywhere?
I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is.
Let $m\geq2$, $n\geq1$ be ...
- 291
8
votes
0
answers
492
views
Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$
Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 99
7
votes
0
answers
141
views
A generalization of matrix minors to non-integer values
I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$
One approach I thought of was to use the fact that the $k$-minors are (...
- 6,741
6
votes
0
answers
392
views
Divisibility properties of minors of matrices
Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ ...
- 191
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
5
votes
0
answers
447
views
Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form
$H_n(a_0(n))=\begin{pmatrix}
a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\
(1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\
(2!)^2 &...
- 51
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
5
votes
0
answers
620
views
Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
4
votes
0
answers
168
views
How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?
I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
- 41
4
votes
0
answers
78
views
Minimal set generators ideal submaximal minors
Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as:
$$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
- 7,300
4
votes
0
answers
96
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 43.2k
4
votes
0
answers
96
views
Are extremal tournament matrices always circulant or 'almost circulant'?
Define an antisymmetric 1-x-matrix as an $n\times n$ matrix $M=(m_{ij})$ with $m_{ii}=0$ and $\{m_{ij},m_{ji}\}=\{1,x\}$ for all $1\le i<j\le n$. Call their set $\mathcal A_n$.
The setup is as ...
- 13.4k
4
votes
0
answers
657
views
determinant of fibonacci-sum graphs
We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...
- 41
3
votes
0
answers
250
views
Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix
The pseudo-determinant of a square matrix $A$ is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma $$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)...
- 141
3
votes
0
answers
295
views
Decomposition of a determinant
Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$.
Does there exist a ...
- 8,998
3
votes
0
answers
104
views
Rank relation to maximum subpermanent and subdeterminant?
Given a $\pm1$ matrix $M$ of rank $r$ let the largest subdeterminant be $d$ and let the largest subpermanent be $p$.
Are there relations/bounds that connect $r$, $d$ and $p$?
Are there geometric and ...
- 13.9k
3
votes
0
answers
184
views
Matrices with only two different entries and maximal determinant
Define $\mathcal M_n$ as the set of all $n\times n$ matrices of full rank with each entry either 1 or $x$.
I am interested in how big the determinant of such a matrix can be. For this, we define in a ...
- 13.4k
2
votes
0
answers
161
views
Jacobi formula for matrices: variations
Jacobi’s formula says: $\frac{d}{dt}\text{det}(A(t))=\text{det}(A(t)) \cdot \text{tr}(\text{Ad}(A(t))\cdot\frac{d}{dt}(A(t))$.
Exists maybe a variation of the Jacobi’s formula where $\text{det}(\frac{...
- 83
2
votes
0
answers
154
views
Applying 1D integral to matrix integral
In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
- 83
2
votes
0
answers
63
views
What is the distribution of determinant of multi multiplication of some Gaussian matrices?
I have a square matric $H = (ABC)(ABC)^H$ where $A$ and $C$ are complex Gaussian matrices with some correlation matrices and $B$ is a diagonal matrix with entries $e^{j \theta}$ on the diagonal such ...
2
votes
0
answers
336
views
For the following class of matrices, are the determinants invariant under permutations?
I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
- 21
2
votes
0
answers
130
views
Pfaffian generalization
The identity
$$\left|
\begin{array}{cccc}
x & y_1 & y_2 & y_3 \\
z_1 & 0 & a & b \\
z_2 & -a & 0 & c \\
z_3 & -b & -c & 0 \\
\end{array}
\right|=\...
- 12.3k
2
votes
0
answers
233
views
Roots of determinant of matrix with polynomial entries — a generalization
For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
- 1,269
2
votes
0
answers
267
views
On the determinant of incidence matrices (of graphs and other geometries)
Let $\Gamma = (P,L,I)$ be a point-line geometry (here, $P$ is the point set, $L$ the line set, and $I$ is the symmetric incidence relation). (As an example, $\Gamma$ could be a graph.) I suppose $\...
- 4,547
2
votes
0
answers
276
views
An (open?) problem about a sequence of nested principal sub-matrices and their determinants
Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
- 59
2
votes
0
answers
179
views
Does this permanent have a closed form?
What is the closed form of this permanent? (similar to the Cauchy determinant)
\begin{aligned}
f(z_1,z_2,\cdots,z_N,w_1,w_2,\cdots,w_N)=\left[
\small{\begin{matrix}
\frac{1}{(z_1-w_1)^2} && \...
- 181
1
vote
0
answers
90
views
When does a matrix with high rank have a minor with disjoint rows and columns and high rank?
This is a somewhat open-ended followup question to
Does an antisymmetric matrix with high rank have a minor with disjoint rows and columns and high rank? and Does a non-singular matrix have a large ...
- 20.2k
1
vote
0
answers
159
views
Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?
Disclaimer: This might be an SE question, but I'm not quite sure...
Thanks in advance!
Setup
So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
- 6,853
1
vote
0
answers
2k
views
Decomposition of Determinant of Sub-Matrices of a Matrix
Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means
$$
{\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...
- 211
1
vote
0
answers
249
views
Is there a way to simplify this apparently huge characteristic polynomial calculation?
Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let $E^{...
- 1,893
1
vote
0
answers
214
views
range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
- 723
0
votes
0
answers
44
views
Special determinant formula
Consider two column vectors $\textbf{a}$ and $\textbf{b}$ of length $k$ and $m$ respectively, $km$ variables denoted $y_{i,j}$ (i=1 to k, j=1 to m), and a quadratic form $\textbf{y}^{T}\mathbb{M}\...
- 419
0
votes
0
answers
51
views
Degree of determinant of a (non-monic) matrix polynomial
Let $n=2, 3, \dots$ and consider the matrix polynomial $L(\lambda)=\sum_{k=0}^{\ell}A_k\lambda^k$, where $A_k \in \mathbb{C}^{n\times n}$.
In the so-called monic case (or that can be made monic by ...
- 11
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
0
votes
0
answers
49
views
Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant
Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
0
votes
0
answers
93
views
Finding the matrix for a given determinant
In my previous question asking about the co-intersection of three circles, a degree six polynomial in twelve variables was found for a special case. This polynomial has precisely 720 terms, of which ...
- 371
0
votes
0
answers
336
views
Pfaffian minors of skew symmetric matrix under perturbation
Suppose $A$ be a skew-symmetric matrix whose entries are positive numbers. A perturbation of $A$, $A'$, is obtained by adding another skew-symmetric matrix whose entries are positive integers.
My ...
- 493
0
votes
0
answers
82
views
The effect of channel error on the determinant of transmitted matrix
Assume the following matrix
$$
E:=\left(
\begin{array}{ccccc}
e_1 & e_2 & \cdots & e_{p-1} & e_{p}\\
e_{p+1} & e_{p+2} & \cdots & e_{2p-1} & e_{2p} \\
\...
- 313
0
votes
0
answers
206
views
Finding a "special" non singular submatrix
Given a square integer matrix $A \in M_n(Z)$ and two subsets $I, J \subset \{ 1, \ldots, n\}$, we define $A_{I,J}$ as the sub-matrix of $A$ containing the rows (resp. columns) whose index is in $I$ (...
- 59