# Questions tagged [skew-symmetric-matrices]

Matrix $A$ is skew-symmetric (or antisymmetric) iff $A^\top = -A$.

176 questions
Filter by
Sorted by
Tagged with
1 vote
35 views

### Is it always true that matrix representation of a skew-symmetric non-degenerate bilinear form is orthogonal with respect to some basis?

I am going through the definition of symplectic matrices. While searching through various properties of those class of matrices I found that every symplectic matrix is necessarily invertible and the ...
• 1,884
42 views

### Manipulation of a multiple cross product expression using skew symmetric matrices

In the derivation of equation (8.23) in the book "Modern Robotics: Mechanics, Planning, and Control" there are some manipulations of the term for the moment contribution of the centripetal ...
42 views

### Find an expression for the nth power of this matrix

I'm looking to derive an expression for $A^n$ Let $A$ = $\begin{pmatrix} 0 & \alpha \\ -\alpha & 0 \end{pmatrix}$ Let $S$ = $\begin{pmatrix} 0 & 1\\ -1& 0 \end{pmatrix}$ Am I correct ...
103 views

### Optimization Problem with a skew-symmetric matrix as a variable

I'm currently trying to solve the following optimization problem $$\mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times n}} \left\| A X - B \right\|_F^2 \quad \text{ subject to } X = -X^T$$ in which ...
101 views

### Intuition for why any square matrix is a sum of a symmetric matrix and a skew-symmetric matrix? [closed]

I've seen proofs for this theorem and understood then. However, I feel like I can gain a better intuition. I would never think of this theorem myself. How could someone discover this theorem? Is ...
• 47
48 views

### Prove that in case of a skew-symmetric matrix, $A$, we have, $A^t=-A$.

I cam accross the following definition in Wikipedia https://en.m.wikipedia.org/wiki/Skew-symmetric_matrix , which is attached below , for a reference: However, I dont get how the two definitions i.e ...
• 205
16 views

1 vote
70 views

### Show that $A + A^{T} = 0$ iff $x^{T}Ax = 0$. [duplicate]

In the first direction, I know that $A + A^{T} = 0$ implies that all diagonal entries must be zero and each non-diagonal has a negative entry, therefore the entire product $x^{T}Ax = 0$ must be equal ...
• 69
117 views

• 130
1 vote
63 views

### Matrix representation of the skew-symmetric operator

Assume that $A$ is a skew-symmetric (skew-hermitian) operator on the finite dimensional unitary vector space $V$. I'm interested in the matrix representation of this operator. I found that there ...
• 517
150 views

• 45
33 views

• 528
121 views

### Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have a very specific question concerning the Fourier transform of certain determinant function of a skew-symmetric matrix: Write $SSym(n)$ for the space of $n\times n$ skew-symmetric real matrices. ...
• 153
39 views

### Modified skew-symmetric matrix eigenvalues

I know that the eigenvalues of a real skew-symmetric matrix $A$ are either zero or imaginary. I have the following modification: let $C$ be a diagonal matrix with ones for some diagonal elements and ...
• 1,741
Let $A = [a_{i,j}]_{i,j} \in M_{2n}(\mathbb{R})$ be a skew-symmetric matrix such that $a_{i,j} \neq 0$, $\forall i \neq j$. I am trying to see if it's true that $\det A \neq 0$. I tried working from ...