All Questions
Tagged with determinants inequalities
16 questions
0
votes
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A conjectured generalization of Oppenheim's inequality, inspired by Horn-Yang's theorem
In this post, $A$ and $B$ are hermitian $n \times n$ positive semidefinite matrices.
It is well known that if $A$ has rank $n$ and if $B$ has only positive entries on its diagonal, then the rank of ...
- 5,215
2
votes
0
answers
80
views
Inequality involving minors of an orthogonal matrix
Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
- 21
8
votes
3
answers
595
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Jensen-like inequality for random matrix: $\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$
Let $X\in M_n(\Bbb R)$ be a random matrix with iid elements following a continuous distribution.
What are the necessary and sufficient conditions for $$\Bbb E[\det X^2]\ge\det\Bbb E[X^2]$$ to hold? Is ...
- 1,459
6
votes
3
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938
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Proof of a matrix implication
If $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.
I have observed by considering many examples of $x,y,z,...
- 51
8
votes
0
answers
492
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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
26
votes
1
answer
1k
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Finding the closest matrix to $\text{SO}_n$ with a given determinant
$\newcommand{\GLp}{\operatorname{GL}_n^+}$
$\newcommand{\SLs}{\operatorname{SL}^s}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\Sig}{\Sigma}$
$\newcommand{\id}{\text{Id}}$
$\newcommand{\...
- 6,741
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
17
votes
2
answers
2k
views
A determinant inequality
Notation: Suppose $\mathbf{A}$ and $\mathbf{B}$ are positive definite matrices in $\mathbb{R}^{n\times n}$ such that $\mathbf{A} \succeq \mathbf{B}$ (Loewner order). Let $\mathcal{S}(n,k)$ be the set ...
- 173
3
votes
1
answer
428
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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 $(\cdot)>0$.
...
- 1,550
12
votes
1
answer
3k
views
Exchange determinant and integral of a matrix-valued function
Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M \det(A)$ and the determinant of its ...
- 195
2
votes
2
answers
383
views
Estimating a Selberg-type integral (or a Fredholm determinant)
I am concerned with the asymptotical behavior of integrals like this for large $n$
$$\frac{1}{n!}\intop_{\Omega}\prod_{1\leq i<j\leq n}(x_{j}-x_{i})^{2}\,\prod_{j=1}^{n}e^{-x_{j}^{2}}dx_{j},$$
...
- 661
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
4
votes
1
answer
278
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Given a correlation matrix $B$. What correlation matrix A (maximizes / minimizes) the following: det(A+B)
Given correlation matrix $B$ (positive semi-definite with ones in the diagonal).
1)Find the correlation matrix $A$ which maximizes $\det\left(A+B\right)$.
2)Find the correlation matrix $A$ which ...
- 41
4
votes
1
answer
829
views
Determinant inequality of square-product sum of diagonal matrix and upper-triangular matrix
Recently, I have seen a matrix inequality but don't know how to prove it. The inequality goes as follows.
For an arbitrary $n\times n$ diagonal matrix $\mathbf{D}$ and an arbitrary upper-triangular ...
- 51
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 $n$-...
- 541
2
votes
3
answers
806
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An Linear Algebra Inequality
How to prove the following inequality:
Let $X$ and $Y$ be $n\times m$ matrices with real entries. Prove that
\begin{equation}
\det\left(XY^T\right)^2 \leq \det\left(XX^T\right)\det\left(YY^T\right) .
\...
- 31