All Questions
103 questions
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
1
answer
102
views
Minimally change matrix with determinant 0
In the following matrix equation, all coefficients $a_{ij}>0$ and all $a_i>0$ and the column sums in the matrix $A$ are all 0
(e.g. $-a_{11}+a_{21}+a_{31}=0$, etc.).
This means that
the ...
- 12.9k
15
votes
3
answers
1k
views
Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 141
10
votes
3
answers
455
views
When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
- 1,724
2
votes
1
answer
345
views
What's the explicit value of this determinant
Let $n\ge2$ be a positive integer, and let $b_1,\cdots,b_n, c_1,\cdots, c_n$ be variables.
Recently, I met the following determinant:
$$\det A=\left|\begin{array}{cccc}
1 & b_1+c_1 & b_1^2+c_1^...
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
5
votes
2
answers
420
views
Maximum determinant of binary matrices with special properties
Let $A$ be an $n$-by-$n$ binary matrix (all elements are in $\{0,1\}$). It is known that the determinant of $A$ is bounded by $O(n^{n/2} / 2^n)$. I am looking for tighter upper bounds for matrices ...
- 3,549
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 1,446
40
votes
6
answers
6k
views
Linear transformation that preserves the determinant
It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on $n\times n$ matrices with complex coefficients that preserves the determinant. Then there exists ...
- 4,713
0
votes
1
answer
525
views
What is the mathematician's definition of the determinant? [closed]
I am trying really hard to find a good definition of the determinant.
I have looked virtually every single resource online and everybody gives a different answer:
sum of cofactors or minors https://...
- 1,300
4
votes
2
answers
2k
views
Determinant of structurally symmetric $n$-banded matrix?
Is there a formula to compute the determinant of a structurally symmetric $n$-banded matrix? I am specifically interested in the 5-banded matrix:
$$ \left[\begin{matrix}
c_{0} & s_{0} & 0 &...
- 367
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 ...
CommunityBot
- 1
6
votes
1
answer
886
views
One question on block-circulant matrices
Circulant matrices are very useful in digital image processing.
I found the general formula for determinant of circulant matrix.
But I think it is not suitable for block-circulant matrices.
For ...
- 52.3k
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 &...
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
2
votes
1
answer
191
views
Monotonicity of the determinant of symmetric Toeplitz Matrices
For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4
\begin{equation}
...
- 1,986
4
votes
2
answers
209
views
Computation of the pfaffian of a particular matrix
This question was originally asked in (https://math.stackexchange.com/questions/4265063/computation-of-the-pfaffian-of-a-particular-matrix). I did not find any satisfactory answer there. So I am ...
- 2,306
1
vote
0
answers
90
views
How to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$? [closed]
Consider a unit norm $\|V\|_2=1$ and a symmetric matrix $A$.
I wish to prove that $\|A^tv\|_2 \leq \|Av\|_2^t$ for every $0<t<1$.
My belief is that this is true is motivated by empirical ...
- 63.9k
4
votes
3
answers
369
views
Determinant in terms of certain $2\times 2$ minors
Let $A$ be an $n\times n$ matrix with entries $a_{i,j}$. Define an $(n-1)\times(n-1)$ matrix $B$ with entries $b_{i,j}=a_{1,1}a_{i+1,j+1}-a_{1,j+1}a_{i+1,1}$. Then $\det(B)=a_{1,1}^{n-2}\det(A)$.
I ...
- 191
16
votes
2
answers
2k
views
Proof that block matrix has determinant $1$
The following real $2 \times 2$ matrix has determinant $1$:
$$\begin{pmatrix}
\sqrt{1+a^2} & a \\
a & \sqrt{1+a^2}
\end{pmatrix}$$
The natural generalisation of this to a real $2 \times 2$ ...
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 $i\in \{1,5 \}$, ...
- 23.7k
3
votes
2
answers
247
views
A problem about determinant and matrix
Suppose $a_{0},a_{1},a_{2}\in\mathbb{Q}$, such that the following determinant is zero, i.e.
$
\left |\begin{array}{cccc}\\
a_{0} &a_{1} & a_{2} \\
\\
a_{2} &a_{0}+a_{1} & a_{1}+a_{...
- 56.9k
8
votes
1
answer
1k
views
Determine if a matrix is unimodular
Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?
- 13.9k
5
votes
2
answers
540
views
How to compute a more general version of Vandermonde / Cauchy double alternant determinant
Consider some variables $\{X_i\}_{1\le i \le n}$, $\{Y_i\}_{1\le i \le n}$, and $\{W_i\}_{1\le i \le n}$. Does anyone know how to compute the following determinant?
$$
\det ~ \left(\frac{W_j^{i-1}}{...
CommunityBot
- 1
2
votes
0
answers
336
views
For the following class of matrices, are the determinants invariant under permutations?
I want to ask a question regarding the invariance of determinants under permutation. The following matrix is the one I want to discuss here. (It's just a symmetric block tridiagonal matrix with non-...
- 21
1
vote
1
answer
546
views
Partial Vandermonde circulant determinant expression
Consider following partial Vandermonde type, circulant matrix
$\begin{bmatrix}
x_1 & x_2 & 0 & \dots & 0 & x_n\\
x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\
\vdots ...
CommunityBot
- 1
2
votes
0
answers
131
views
Pfaffian generalization
The identity
$$\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
5
votes
1
answer
335
views
Determinant of a certain Toeplitz matrix
Compute the following determinant
\begin{vmatrix} x & 1 & 2 & 3 & \cdots & n-1 & n\\ 1 & x & 1 & 2 & \cdots & n-2 & n-1\\ 2 & 1 & x & 1 &...
- 34.3k
53
votes
7
answers
51k
views
Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-...
- 11
6
votes
1
answer
194
views
Values of a pair of determinants
Let $\mathbf{x} = (x_0, x_1, x_2), \mathbf{y} = (y_0, y_1, y_2)$ be vectors over a field $\mathbb{F}$ of characteristic zero. Define the function
$\displaystyle S(\mathbf{x}, \mathbf{y}) = x_2 (y_0^2 -...
- 3,984
1
vote
1
answer
254
views
When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?
Let $A$ be an $n \times n$ real symmetric matrix.
Let
$$
M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix}
$$
where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...
7
votes
1
answer
1k
views
Block matrices and their determinants
For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows:
(a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...
5
votes
2
answers
4k
views
Determinant of block tridiagonal matrices
Is there a formula to compute the determinant of block tridiagonal matrices when the determinants of the involved matrices are known?
In particular, I am interested in the case
$$A = \begin{pmatrix} ...
63
votes
7
answers
9k
views
How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
15
votes
3
answers
6k
views
Questions on Toeplitz matrices: invertibility, determinant, positive-definiteness
These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in Math Stack Exchange.
Let $A$ be an $n \times n$ Hermitian Toeplitz matrix:
$$A = \begin{...
6
votes
2
answers
340
views
Characteristic polynomial of checker matrix
For every integer $n > 0$, let $C_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $...
- 2,720
16
votes
2
answers
2k
views
How to prove the determinant of a Hilbert-like matrix with parameter is non-zero
Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\...
- 243
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
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
$$ [H_n]_{ij} = \frac{...
- 51
4
votes
1
answer
1k
views
Determinant involving traceless unitary hermitian matrices
Let $S$ be the set of complex $N\times N$ matrices that are traceless, unitary and hermitian.
A friend asked me the following question, motivated by a problem in condensed matter physics:
Is it ...
- 1,593
10
votes
0
answers
237
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 4,729
13
votes
2
answers
1k
views
Determinant of $V^* V$ where $V$ is rectangular Vandermonde matrix with nodes on unit circle
Let $z_{1},\dots,z_{k}$
be distinct complex numbers with $\left|z_{j}\right|=1,\;j=1,\dots,k$. For any natural $N\geqslant k$
consider the rectangular Vandermonde matrix
$$
V_{N}=\begin{pmatrix}1 &...
- 3,716
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\...
- 356
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 $P$-...
CommunityBot
- 1
11
votes
4
answers
5k
views
Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices
What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?
- 63.9k
31
votes
1
answer
4k
views
Determinants of binary matrices
I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
- 12.7k
28
votes
4
answers
5k
views
Jacobi's equality between complementary minors of inverse matrices
What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...
- 33.8k
5
votes
2
answers
312
views
minimum-maximum entries matrix
Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries
$$M_{i,j}(n)=\frac{x_{\max(i,j)}}{x_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$
I'm interested in the following:
...
CommunityBot
- 1
9
votes
3
answers
409
views
Determinant of a block matrix with many $-1$'s
For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...
- 23.7k