All Questions
103 questions
7
votes
3
answers
1k
views
Determinant of correlation matrix of autoregressive model
I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form
$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-...
- 43.2k
7
votes
1
answer
248
views
The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 109k
16
votes
0
answers
809
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 489
4
votes
1
answer
296
views
Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
CommunityBot
- 1
1
vote
0
answers
2k
views
Decomposition of Determinant of Sub-Matrices of a Matrix
Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means
$$
{\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...
- 211
6
votes
2
answers
738
views
Probability of a large random integer Matrix to have zero determinant
Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$
I would like to ...
CommunityBot
- 1
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?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
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 ...
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
-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 \...
- 20.2k
12
votes
1
answer
624
views
Determinants: periodic entries $0,1,2,3$
Consider an $n\times n$ matrix $M_n$ where the sequence
$$\{1,2,3,\dots,n^2\} \mod 4=\{1,2,3,0,1,2,3,\dots\}$$ forms a clock-wise spiral, in that given order. For example,
$$M_4=\begin{bmatrix} 1&...
- 33.8k
7
votes
4
answers
1k
views
Generalized Cauchy-Binet sum over a fixed subset of indices
I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
- 13.6k
5
votes
0
answers
315
views
Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.9k
8
votes
1
answer
726
views
A direct proof of a property of symmetric 2x2-determinants
Let $f(a,b,c)=\det\begin{pmatrix}a &b\\ b& c\end{pmatrix}\in\mathbb{R}[a,b,c]$ be the determinant of a $2 \times 2$ real symmetric matrix.
Let $f(x_i,y_i,z_i)\geq 0$, $x_i\geq 0$, $z_i\geq 0$ ...
- 14k
15
votes
1
answer
8k
views
On the determinant of a class symmetric matrices
Consider the matrix $2\times2$ symmetric matrix:
$$
A_2=\begin{pmatrix} 1 & a_1 \\ a_1 & 1\end{pmatrix}.
$$
It's clear that the restriction $|a_1|<1$ implies that $\det(A_2)>0$. Moreover,...
- 109k
6
votes
0
answers
375
views
Monomial base change and the Vandermonde
Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$.
The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$.
It is well-known that in as much as ...
- 43.2k
4
votes
1
answer
183
views
dyadically recursive matrices: Part II
We present some variation on this MO question which is equally amusing to me.
Define the $2^{n-1}\times 2^{n-1}$ matrix $A_n$ recursively as follows: $A_1(a_1)=\begin{pmatrix} a_1\end{pmatrix}$ and
$$...
CommunityBot
- 1
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
CommunityBot
- 1
8
votes
4
answers
3k
views
Some Hankel Determinants
After invoking a recursion relation for Hankel determinants in my answer to a (mostly unrelated) question, I started wondering what else I could use this recursion for, and stumbled upon some results ...
- 43.2k
14
votes
2
answers
873
views
"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
- 63.9k
8
votes
1
answer
321
views
"Almost Hankelized" numerical Vandermonde
One of the more utilized determinant is that of Vandermonde's
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
1&x_2&x_2^2&\dots&x_2^{n-1}\\
\ldots&\ldots&\...
- 85.6k
5
votes
2
answers
335
views
Determinant of the "quantum" version of the group $\mathbb{Z}_n$
Let $[0]_q:=0$ and $[n]_q:=\frac{1-q^n}{1-q}=1+q+\cdots+q^{n-1}$, for $n\geq1$.
Question. Is there a closed formula (with proof) for the determinant of the matrix of $(i,j)$-entries
$$[i+j\bmod n]...
- 43.2k
11
votes
3
answers
918
views
yet another determinant and inverse of a matrix
This problem is some variation of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}-c& a & a& \dots & a \\ b & c & a& \ddots & a\\ b & b & -c &...
CommunityBot
- 1
5
votes
2
answers
203
views
dyadically recursive matrices: Part I
Introduce the $2^{n-1}\times 2^{n-1}$ matrix $B_n$ recursively as follows: $B_1(b_1)=\begin{pmatrix} b_1\end{pmatrix}$ and
$$B_n(b_1,\dots,b_n)=\begin{pmatrix} B_{n-1}(b_1,\dots,b_{n-1})& b_nJ_{n-...
- 43.2k
1
vote
0
answers
143
views
How to show the determinant of $B - I$ is zero? [closed]
Let $n \geq 2$ be a positive integer and
$$\beta_i= \left(
\begin{array}{c|c c|c}
I_{i-1} & 0 & 0 & 0\\
\hline
0 & 1-q & q & 0\\
0 & 1 & 0 & 0\\
\hline
...
15
votes
3
answers
3k
views
Determinant of a $k \times k$ block matrix
Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...
18
votes
3
answers
6k
views
Number of unique determinants for an NxN (0,1)-matrix
I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
0
votes
0
answers
336
views
Pfaffian minors of skew symmetric matrix under perturbation
Suppose $A$ be a skew-symmetric matrix whose entries are positive numbers. A perturbation of $A$, $A'$, is obtained by adding another skew-symmetric matrix whose entries are positive integers.
My ...
- 493
10
votes
1
answer
520
views
Homogeneous polynomials, mixed determinants, positive definiteness
Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial
$$
f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n})
$$
never vanishes on $\...
- 109k
8
votes
2
answers
796
views
generalizations of Vandermonde matrix to high dimensions
Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix
the maps
$$
f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$
x\longmapsto (1,x,x^2,\cdots,x^{n-1})...
- 12.9k
3
votes
0
answers
915
views
How to find a closed form of following matrix's determinant [closed]
I wanna find a closed form of determinant of the following matrix
$$A(n) =
\begin{pmatrix}
B_{1} & B_{2} & \cdots & B_{n} & 1 \\
B_{n} & B_{1} & \cdots & B_{n-1} &...
- 513
11
votes
2
answers
964
views
How to prove this determinant is positive-II?
Question: Given an arbitrary number of real matrices of the form $ A_i=
\biggl(\begin{matrix}
C_i+E_i & B_i \\
B_i^T & D_i-F_i
\end{matrix} \biggr)
$, where $B_i$ is an arbitrary $n\times n$ ...
CommunityBot
- 1
3
votes
1
answer
428
views
Inverse Hadamard determinant inequality
As far as I remembered there is an inverse Hadamard inequality for the determinant of the form
$$
|D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)}
$$
providing all values in $(\cdot)>0$.
...
- 2,286
11
votes
0
answers
1k
views
How the idea of adjugate matrix has been designed? [closed]
I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...
- 27.9k
3
votes
1
answer
559
views
Determinant of a Certain Positive-Definite Block Matrix
Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...
CommunityBot
- 1
4
votes
5
answers
4k
views
About adding a negative definite rank-1 matrix to a symmetric matrix
If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$)
I guess that the eigenvalues of $B - vv^T$ ...
- 1,748
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
10
votes
1
answer
938
views
Why does this antisymmetric product factor out a determinant?
Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...
- 33.8k
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
6
votes
1
answer
954
views
Proving that the kernel of this matrix is of dimension 2
(Edit : see at the bottom of the question for an additional surprising possible hint.)
Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when $...
CommunityBot
- 1
0
votes
1
answer
204
views
Are these particular kinds of matrices well known?
Given two positive integers $n$ and $a \leq \frac{n}{2}$ consider a $n \times n$ matrix $A$ such that,
all the diagonal entries are either $a$ or $a+1$
all the non-zero off-diagonal entries are $\pm ...
- 9,650
2
votes
1
answer
221
views
Are there good ways of relating a minor to the full determinant?
Say $A$ is a $(n-1)\times (n-1)$ matrix and we augment it by a $n^{th}$ row and a column and get a $n \times n$ matrix $B$. Is there a nice way to relate $det(B)$ and $det(A)$ and the added row and ...
- 52.3k
1
vote
0
answers
108
views
MInors related problem [closed]
A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is $||x||=\sqrt{x_1^2+x_2^2+...
- 11
3
votes
1
answer
320
views
Number of Matrices with bounded determinant
Here's my question:
Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
CommunityBot
- 1
10
votes
1
answer
836
views
Factor a sum of products of cofactors
Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...
CommunityBot
- 1
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 ...
- 424
2
votes
3
answers
755
views
On matrices in linear forms with vanishing determinant
This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.
Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...
CommunityBot
- 1
5
votes
2
answers
2k
views
Determinant of non-symmetric sum of matrices
Given three real, symmetric matrices $A\succ0$ and $B$, $C⪰ 0$.
How can it be shown that:
$$\det(A^2+AB+AC) \leq \det(A^2 +BA +AC+BC) ? \qquad (\star)$$
Where $A^2$ is symmetric and positive ...
- 223
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 ...
CommunityBot
- 1
5
votes
4
answers
8k
views
Proving a determinant = 0
The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...
- 51