All Questions

Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...

If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B. Given then all these 2x2 determinants of an ...

Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix. It is claimed that $$ \|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...

Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that \begin{align} F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N. \end{align} What we can say about the singular value ...

For given $n\times n$ real symmetric positive definite matrices $X_{1}, X_{2}$, let $Y := X_{1}^{-1}X_{2}$, and let $I$ be the identity matrix. I would like to solve the following equation for the ...