Questions tagged [skew-symmetric-matrices]

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

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Dimensions of symmetric and skew-symmetric matrices

Let $\textbf A$ denote the space of symmetric $(n\times n)$ matrices over the field $\mathbb K$, and $\textbf B$ the space of skew-symmetric $(n\times n)$ matrices over the field $\mathbb K$. Then $\dim (\textbf A)=n(n+1)/2$ and $\dim (\textbf…
34
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Why are all nonzero eigenvalues of the skew-symmetric real matrices pure imaginary?

Assume that $A$ is an $n\times n$ skew-symmetric real matrix, i.e. $$A^T=-A.$$ Since $\det(A-\lambda I)=\det(A^T-\lambda I)$, $A$ and $A^T$ have the same eigenvalues. On the other hand, $A^T$ and $-A$ also have the same eigenvalues. Thus if…
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Proof that the rank of a skew-symmetric matrix is at least $2$

Is there a succinct proof for the fact that the rank of a non-zero skew-symmetric matrix ($A = -A^T$) is at least 2? I can think of a proof by contradiction: Assume rank is 1. Then you express all other rows as multiple of the first row. Using…
Naga
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Matrix exponential of a skew-symmetric matrix without series expansion

I have the following skew-symmetric matrix $$C = \begin{bmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{bmatrix}$$ How do I compute $e^{C}$ without resorting to the series expansion…
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Is it possible to have a $3 \times 3$ matrix that is both orthogonal and skew-symmetric?

Is it possible to have a $3 \times 3$ matrix that is both orthogonal and skew-symmetric? I know it has something to do with the odd order of the matrix and it is not possible to have such a matrix. But what is the reason?
16
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Existence of the Pfaffian?

Consider a square skew-symmetric $n\times n$ matrix $A$. We know that $\det(A)=\det(A^T)=(-1)^n\det(A)$, so if $n$ is odd, the determinant vanishes. If $n$ is even, my book claims that the determinant is the square of a polynomial function of the…
Potato
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Determinant of a real skew-symmetric matrix is square of an integer

Let $A$ be a real skew-symmetric matrix with integer entries. Show that $\operatorname{det}{A}$ is square of an integer. Here is my idea: If $A$ is skew-symmetric matrix of odd order, then $\operatorname{det}{A}$ is zero. So, take $A$ to be of…
user51266
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4 answers

Why are skew-symmetric matrices of interest?

I am currently following a course on nonlinear algebra (topics include varieties, elimination, linear spaces, grassmannians etc.). Especially in the exercises we work a lot with skew-symmetric matrices, however, I do not yet understand why they are…
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Problem in skew-symmetric matrix

Let $A$ be a real skew-symmetric matrix. Prove that $I+A$ is non-singular, where $I$ is the identity matrix.
user12290
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Equivalence of skew-symmetric matrices

Let $N=\{1,\dots,n\}$ and $A,B$ be $n\times n$ skew symmetric matrices such that it is possible to permute some rows and some columns from $A$ to get $B$. In other words, for some permutations $g,h: N\rightarrow N$, $$A_{i,j}=B_{g(i),h(j)}$$ for…
Karo
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Let $A,B\in M_3(\mathbb{C})$ be invertible matrices such that $AB=BA=X$, $A^{T}+A=B^{T}+B=X^{T}+X$. Then, is $\det(X - I) = 0$?

Let $A,B\in M_3(\mathbb{C})$ be invertible matrices such that $AB=BA=X$, $A^{T}+A=B^{T}+B=X^{T}+X$, then: (A) $A=B$ (B) $\det(A-I)=0$ (C) $\det(B-I)=0$ (D) $\det(X-I)=0$ My working: $AB+(AB)^T=X+X^T\implies AB+B^TA^T=B+B^T\implies…
Makar
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The squares of skew-symmetric matrices span all symmetric matrices

This is a self-answered question. I post this here, since it wasn't obvious for me at first, and I think it might be helpful for someone at some future time (maybe even future me...). Claim: Let $n \ge 3$, and let $X$ be the set of all squares of…
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Every skew-symmetric matrix has a non-negative determinant

Let $A$ be a skew-symmetric $n\times n$-matrix over the real numbers. Show that $\det A$ is nonnegative. I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ skew-symmetric matrix). So when $n$ is odd, we…
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2 answers

How do I find $\operatorname{det} T_Q$?

Let $S$ be the space of all $n \times n$ real skew symmetric matrices and let $Q$ be a real orthogonal matrix. Consider the map $T_Q: S \to S$ defined by $$T_Q(X) = QXQ^T.$$ Find $\operatorname{det} T_Q$. I thought about diagonalizing $Q$, but I…
5
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Matrix exponential of the sum of two skew-symmetric matrices

This is my first message in this site. I'm a mechanical engineer with, amongst others, an interest in inertial navigation. I'm currently reading the book "Principles of GNSS, Inertial and Multisensor Integrated Navigation Systems" from Paul D.…
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