Articles of matrices

Degree $2$ nilpotent matrices with non-zero product

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 […]

Multiplication of Rational Matrices

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. […]

Why one would want to normalize a matrix by dividing it by its Frobenius norm?

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 […]

system of equation (3 unknown, 3 equations)

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 $x=\frac a{3-a}\implies (a+1)\frac a{3-a}+z(1-a)=1\implies$ […]

Periodicity of $\log(\exp(M))$

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= […]

Methods to solve a system of many Ax=B equations using least-squares

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 […]

Positive semidefinite but non diagonalizable real matrix – proof real parts of eigenvalues are non-negative

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 […]

Set of all positive definite matrices with off diagonal elements negative

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. […]

Vector $p$-norm for square matrices is submultiplicative for $1 \le p \le 2$

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 […]

Prove that $A$ is similar to $A^n$ based on A's Jordan form

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 […]