All Questions
Tagged with determinants matrix-theory
20 questions
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
2
votes
1
answer
137
views
Existence of matrices with some invertibility properties
Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$.
I am interested ...
user83947
1
vote
1
answer
187
views
Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties
All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$ and let $e_1$, $e_2$, $\ldots$, $e_{10}$ denote its rows. For $i\in \{1,5 \}$, ...
user83947
2
votes
1
answer
81
views
Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties
All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which
there exists a matrix $X$ ...
user83947
6
votes
0
answers
584
views
Expressing a polynomial as the determinant of a matrix of linear forms
I have heard that it's a well known result (in theoretical computer science?) that if we have a polynomial $p(t_1,\dots,t_n)$ over $\mathbb Q$, we can find matrices $M_0,\dots,M_n/\mathbb Q$ such that ...
- 7,746
9
votes
2
answers
515
views
Minors of low rank skew-symmetric matrix
Let $A$ be an $n\times n$ skew-symmetric matrix of rank $r$.
Given subsets $X$ and $Y$ of row and column indices respectively, let $A_{X,Y}$ denote the submatrix of $A$ obtained by only keeping rows ...
- 557
6
votes
0
answers
203
views
Conjecture for a certain Cauchy-type determinant
Given the Cauchy-like matrix
$$
\mathbf X_M(q) = \left[ \frac{2}{\pi} \frac{
\Gamma\!\left(m - \frac{1}{2} \right)\Gamma\!\left(n + \frac{1}{2} \right)
}{
\Gamma(m)\,\Gamma(n)
}
\frac{m-\frac{3}{4}}
{\...
- 3,691
7
votes
1
answer
248
views
The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 1,997
16
votes
5
answers
2k
views
Expected value of determinant of simple infinite random matrix
Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$
I would like to ...
- 355
0
votes
1
answer
77
views
$A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$
Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m ...
- 39
2
votes
1
answer
111
views
Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$
Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n}$. $D$ is a positive diagonal matrix and $m > n$.
Is there any equality or inequality over $|B|$, $|AA^...
- 39
-1
votes
1
answer
195
views
Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 2,246
7
votes
1
answer
3k
views
When does the determinant distribute over addition?
When does $\det(A+B)=\det(A)+\det(B)$ hold?
I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
- 13.9k
6
votes
0
answers
349
views
Exact determinant of a Cauchy-like matrix
Question. Is there a closed-form expression for the determinant of a $n \times n$ matrix $A$ with entries
$$
A_{i,j} = \frac{1 - \delta_{i, j}}{z_i - z_j}, \qquad 1\leq i, j\leq n,
$$
where $z_i$ ...
- 263
2
votes
1
answer
257
views
How to characterize singular matrix $X$ that solves det$(X−A)=0$, where $A$ is symmetric positive definite?
Consider real square matrices $X$ and $A$ of same size, where $A$ is known to be symmetric positive definite. I came across the matrix equation $XX^{\top} = AX^{\top}$, which solved for $X$ gives ...
- 1,538
28
votes
2
answers
16k
views
Determinants in Graph Theory
In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various linear-algebraic properties. For example, their trace can be calculated (...
- 381
1
vote
1
answer
18k
views
Derivative of log determinant and inverse
I have a matrix $\Sigma$ with element $(i,j)$
$$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$
The matrix is positive definite and symmetric (it is a covariance matrix).
Now I need to evaluate
$$\frac{\...
- 29
5
votes
2
answers
1k
views
Generalizations of Oppenheim's inequality
The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$.
There has been a lot of beautiful work done ...
- 6,962
20
votes
1
answer
25k
views
When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?
This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close.
Here is the motivation: If you have ever taught a maths course for engineers ...
- 12.7k
22
votes
2
answers
14k
views
Infinite matrices and the concept of "determinant"
Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
- 1,861