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16 votes
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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 ...
Krokop's user avatar
  • 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\...
Daniil Rudenko's user avatar
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 $...
Jiro's user avatar
  • 909
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? $$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
Martin's user avatar
  • 99
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 $d_i(A)$ ...
Joel Louwsma's user avatar
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 ...
T. Amdeberhan's user avatar
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 &...
fs98's user avatar
  • 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 $\det M'=(a'+b')(c'+d')$...
Turbo's user avatar
  • 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? ...
user6818's user avatar
  • 1,893
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 ...
Puzzled's user avatar
  • 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 ...
Turbo's user avatar
  • 13.9k
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-...
Charles Cao's user avatar
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|=\...
Alexey Ustinov's user avatar
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 ...
H A Helfgott's user avatar
  • 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 $P$-...
dohmatob's user avatar
  • 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\, ...
user0410's user avatar
  • 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^{...
user6818's user avatar
  • 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 ...
Paul B. Slater's user avatar
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 ...
94thomas's user avatar
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 ...
ABB's user avatar
  • 4,058
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 ...
SiOn's user avatar
  • 493