Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and (2) every product of the permutations of $M_i$ doesn’t vanish: $\prod_{i} M_i \ne 0$? If it is possible, $M_i$ better be as small […]

Let $\mathbf A(x)$ and $\mathbf B(x)$ be $n \times n$ rational matrices, whose elements are rational functions in the scalar $x \in \mathbb R$. Suppose that $\mathbf A(x) \mathbf B(x)$ is a polynomial matrix in $x$, meaning that the denominators in the elements of $\mathbf A(x)$ and $\mathbf B(x)$ somehow cancel out in the multiplication. […]

I am currently reading a scientific paper about clustering of brain signals, which consist on long time series across many channels (each signal is a matrix of C channels by T time samples). In the preprocessing of their datas, the authors normalize each signal matrix by dividing it with its Frobenius norm. My problem is […]

I want to solve a system of equations, but I seem to get it wrong. Problem: see picture, and note that I should tell for which a I have no solution one solution $\infty$ solutions Attempt: $$\begin{matrix}i \\ ii \\ iii\end{matrix}\left\{\begin{matrix}x+y-az=3\\ ax-y+z=-2\\ -3x+y-z=-a+2\end{matrix}\right.$$ $ii+iii\to (a-3)x=-a\implies x=-\frac a{a-3}=\frac a{3-a}$ $i+ii\to (a+1)x+z(1-a)=1$ and $x=\frac a{3-a}\implies (a+1)\frac a{3-a}+z(1-a)=1\implies$ […]

Let $m_3\in so(3)$ and say $$ m_3= \begin{bmatrix} 0 & -6 & 5 \\ 6 & 0 & -4 \\ -5 & 4 & 0 \end{bmatrix} $$ It is well known that $\exp$ is many-to-one, hence if $n_3=\log(\exp(m_3))$, then $n_3\ne m_3$. Now let $\overline m_3$ be the compact representation of $m_3$, hence $$ \overline m_3= […]

I am working with a force measurement instrument which needs calibration via a calibration matrix. For each of a set of controlled measurements I have a vector $k$ of three known, independent values, and a corresponding vector $m$ containing three measured values, that due to instrument constraints end up being non-orthogonal. From each measurement, I […]

I have a question about positive semidefinite matrices that are non diagonalizable. Example: \begin{equation} A= \left(\begin{array}{cc} 2 & 1\\ 0 & 2\\ \end{array}\right) \end{equation} Clearly the (real part of the) eigenvalues of $A$ are non-negative. But how do I prove in general that the real part of the Eigenvalues of a positive semi-definite real matrix […]

Please, help me to find the set of all positive definite matrices (PDM) of which off diagonal elements are negative. Considering the case with n=2, the symmetric mat $A=[a_1, a_2;a_2, a_3]$ needs to be a PD. i.e. $x’Ax>0$, $a_2\neq 0$, $x\neq 0$ where $x=[x_1,x_2]’$. Always we have $a_1>0$ and $a_3>0$ as they are principal minors. […]

I’m trying to prove that the vector $p$-norm for square matrices is submultiplicative for values of $p$ between $1$ and $2$. The vector $p$-norm for a square matrix $A$ is defined as $\displaystyle \|A\|_p:=\left(\sum_{i=1}^n \sum_{j=1}^n |a_{ij}|^p \right)^{\frac{1}{p}}$ For $p=1$ and $p=2$ the result follows easily from the Cauchy-Schwarz inequality. Now for $1<p<2$ my idea is […]

Let $A = \begin{bmatrix}1&-3&0&3\\-2&-6&0&13\\0&-3&1&3\\-1&-4&0&8\end{bmatrix}$, Prove that $A$ is similar to $A^n$ for each $n>0$. I found that the characteristic polynomial of $A$ is $(t-1)^4$, and the minimal polynomial is $(t-1)^3$. And the Jordan form of $A$ is \begin{bmatrix}1&1&0&0\\0&1&1&0\\0&0&1&0\\0&0&0&1\end{bmatrix} I guess the key to solve this is to use the fact that two matrices are similar […]

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