Questions tagged [skew-symmetric-matrices]

Question 1

Prove that the determinant of a skew-symmetric matrix of even order does not change if to all its elements we add the same number. i tried calculating $\det(A+cB)$,$c \in R$, $(b)_{ij}=1 \forall i,j$ ...

Question 2

I am reading Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III. On p.16 in this book: Exercise 2.5. Let $S\in\mathbb{C}^{m\times m}$ be skew-hermitian, i.e., $S^*=-S$. (a) Show by ...

Question 3

We define $A$ to be a diagonal matrix with diagonal entries: one $p_1$ and $2n-1$ many $p_2$, here both $p_1$ and $p_2$ are real and $p_1p_2>-1$. $J$ is a skew-symmetric matrix of form $$J= ...

Question 4

I am new to lie theory and representation theory. I heard about this interesting factorization known as the Bipolar decomposition which uses the Mostow decomposition. The article is https://www....

Question 5

I know that for any skew-symmetric matrix $A$, the exponential $e^A$ is an orthogonal matrix due to the identity: $(e^A)^T = e^{-A} = (e^A)^{-1}$ However, I’m looking for a deeper geometric intuition: ...

Question 6

Consider the space of skew-symmetric matrices skew$(p)$, where $A = -A^T$ when $A \in \text{skew}(p)$. Let $S_p$ be the group of $p-$permutation matrices. $S_p$ acts on $\text{skew}(p)$ via ...

Question 7

Is it possible to create a rank 3 tensor that is skew-symmetric in one pair of indices but symmetric in another pair? That is, a tensor whose components satisfy $$S_{ijk}=-S_{jik}=S_{kji}$$ How many ...

Question 8

I have a matrix differential equation $\dot{A}(t) = A(t)B(t)$ where $A$ is an orthonormal matrix and $B$ is a known skew-symmetric matrix, both 3-by-3. The farthest I could get in finding a general ...

Question 9

If $Y$ is a column matrix and $X$ a square matrix such that $Y^{\top}XY=0$ for all $Y$, then prove that $X$ is skew symmetric. (They didn't specify the field, so I'm going to assume its the real ...

Question 10

I have a skew-symmetric $n\times n$ matrix $M$ such that all its upper-triangular entries are $1$: $$ M_{i,j} = \begin{cases} \,\,\, 1 & \text{ if } i < j \\ \,\,\, 0 & \text{ if } i = j \...

Question 11

Is it true that, if we take an antisymmetric matrix $A$, $A^\top = -A$, of dimension $n \times n$ and we parametrize it as $$ A = \begin{pmatrix} 0 & a_{12} & -a_{13} & a_{14} & \cdots ...

Question 12

I am trying to verify the derivation of the following expression involving a skew-symmetric matrix $ [\mathbf{u}_3]_\times $, a symmetric matrix $ \boldsymbol{\omega}^* $ representing a cross-product ...

Question 13

I have difficulty in solving the following problem (P4.2.3 in 《Matrix Computations》) Suppose $A\in\mathbb{R}^{n\times n}$ be positive definite, and set T = (A+$A^T$)/2, show that $||A^{-1}||_2\leq ||T^...

Question 14

Consider the matrix $(Id - JQ)$, where $Id \in \mathbb{R}^{n \times n}$ is the identity matrix, $J^T = - J \in \mathbb{R}^{n \times n}$ is a skew-symmetric matrix and $\mathbb{R}^{n \times n} \ni Q^T =...

Question 15

Let $A$ be a random matrix, which elements take values $1$ and $0$ with probably $p$ and $(1-p)$ respectively. What is $\mathbb{E}[det(A-A^T)]$? When dimensionality of $A$ is odd, $det(A-A^T) = 0$, as ...

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Questions tagged [skew-symmetric-matrices]

A quest about the determinant of a skew symmetric matrix [duplicate]

How to use Exercise 2.1 to solve Exercise 2.5(a)? (Numerical Linear Algebra by Lloyd N. Trefethen and David Bau, III.)

Determinant (eigenvalues) of the difference of a diagonal matrix and a skew-symmetric matrix.

Geometric interpretation of the matrix exponential of imaginary symmetric and imaginary skew symmetric matrices?

Can $e^A$ be visualized as a rotation in $\mathbb{R}^3$ when $A$ is a $3 \times 3$ real skew-symmetric matrix?

Representation of $S_n$ over skew symmetric matrices skew$(n)$

Rank-3 tensor that's skew and symmetric on different pairs of indices

How to solve $\dot{A}(t)=A(t)B(t)$

Proving $X$ is a skew-symmetric matrix given $Y^{\top}XY=0$

Proving a relation among powers of a $\pm 1$ skew-symmetric matrix

Trace of the square of an antisymmetric matrix

Deriving the Equality for $ \mathbf{U}^\top [\mathbf{u}_3]_\times \boldsymbol{\omega}^* [\mathbf{u}_3]_\times \mathbf{U} $

Norm inequality about positive definite matrix, along with its symmetric form and skew symmetric form

Is the matrix $(Id - JQ)$ regular for any skew-symmetric matrix $J$ and symmetric positive semi-definite matrix $Q$? [duplicate]

Find $\mathbb{E}[\det(A-A^T)]$ if $a_{ij} \sim \text{Bernoulli}(p)$ [closed]

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