Skip to main content

[Community Question] Linear-algebra: Number of possible zero entries in orthogonal matrices

One of our user asked:

It's easy to check that in orthogonal matrix dimension $2 \times 2$ if there is entry $0$ in the matrix necessary one additional zero must be present. Then the total number of zeros is $2$.

In an orthogonal matrix dim. $3 \times 3$ number of zeros can be (if they are present) , I suppose from observations, only $4$ or $6$ - once again we obtain an even number of possible zeros.

  • Can this observation be extended for other orthogonal matrices of greater dimensions? The number of zeros is always even? How to prove this?

  • Maybe, it is known the explicit formula for the number of possible zeros in orthogonal matrices of any dimension?


Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Linear-algebra: Are linear transformations between infinite dimensional vector spaces always differentiable?

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).
Post a Comment
Read more

[Community Question] Calculus: prove $\int_0^\infty \frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}{8}$ with real methods

One of our user asked: I am attempting to prove that $$J=\int_0^\infty\frac{\log^2(x)}{x^2+1}\mathrm dx=\frac{\pi^3}8$$ With real methods because I do not know complex analysis. I have started with the substitution $x=\tan u$ : $$J=\int_0^{\pi/2}\log^2(\tan x)\mathrm dx$$ $$J=\int_0^{\pi/2}\log^2(\cos x)\mathrm dx-2\int_{0}^{\pi/2}\log(\cos x)\log(\sin x)\mathrm dx+\int_0^{\pi/2}\log^2(\sin x)\mathrm dx$$ But frankly, this is basically worse. Could I have some help? Thanks.
Post a Comment
Read more