10 questions linked to/from Rank of skew-symmetric matrix
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Every skew-symmetric matrix has even rank [duplicate]

Let $F$ be a field where $char(F)\neq2$ and let $A$ be a skew-symmetric matrix over $F$. Prove that rank of $A$ is even. I think the best way to prove it, is using induction on size of $A$. for $n=1$...
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Prove that the rank of a real skew symmetric matrix is not $1$

I was thinking of using the RREF of $A$, where $A$ is an $n \times n$ skew symmetric matrix. So, suppose that it's rank is 1. then the first row has at least one non zero element(which is 1), while ...
900 views

Show that rank of skew-symetric is even number

$$A = -A^T$$ I assume that $A$ is not singular. So $$\det{A} \neq 0$$ Then $$\det(A) = \det(-A^T) = \det(-I_{n} A^T) = (-1)^n\det(A^T) = (-1)^n\det(A)$$ So I get that $n$ must be even. But what ...
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Proof Verification: Skew-symmetric matrix has even rank

I thought of this argument to show that the rank of a skew-symmetric matrix is even. I'd appreciate it if someone could look over it and let me know whether or not it looks correct. Let $A$ be an $n$...
154 views

Finding rank of matrix

Suppose $B$ is a non-zero real skew-symmetric matrix of order $3$ and $A$ is a non-singular matrix with inverse $C$. Then rank of $ABC$ is: (A) $0, 1, 2$ (B) definitely $1$ (C) definitely $2$ (D) ...
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Let $F$ be a field such that 1+1 is not equal to zero. Let $A$ element of $M_n(F)$ such that $A^T = -A$.

Let $F$ be a field such that 1+1 is not equal to zero. Let $A$ element of $M_n(F)$ such that $A^T = -A$. a) Show that if $A$ is non-singular, then $n$ is even. b) Show that if $A$ is singular, ...
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$vv^TA-Avv^T$ where $A$ is skew-symmetric and $vv^T$ is rank one

Suppose I have the following: $$vv^TA-Avv^T$$ $v\in \mathbb{R}^{n}$ with $\|v\|_2=1$, so $vv^T$ is PSD, rank one and $\operatorname{tr}(vv^T)=1$ $A$ is skew-symmetric Is there any nice ...
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how is the rank of a skew map is even ?

In the book of Linear Algebra by Greub, at page 230, it is claimed that More general, it will now be shown that the rank of a skew transformation is always even. Since every skew mapping is ...
Rank of a matrix $A$ such that $A + A^T = 0$
I need to prove (using only elementary operations and induction) that rank of a matrix $A\in \Bbb C^{n\times n}$ such that $A + A^T = 0$ is an even number. I know that elementary operations doesn't ...
Consider a convex polygon inscribed in a circle with vertices $P_1$, ..., $P_n, \ n \ge 3$. Let $A$ be the matrix $n \times n$ such that \$\begin{equation} a_{ij} = \begin{cases} |P_iP_j| ...