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I need to create a series of test matrices to unit test a Python function I have to compute the rank of a matrix in $GF(2)$. This is difficult for me because I don't have any other Python function tha …

Given $u,v,w,x,y,z$ is there a way to solve the following for $a,b,c$, assuming that there is a solution (e.g. $(u,v,w)$ and $(x,y,z)$ are not proportional): $$\begin{aligned} au+bv &= cw\\ ax+by &= c …

I have a matrix problem I don't know how to solve because I can't seem to express it as a multiplication of matrices. Suppose I have the following 4x4 matrix $A$: $$\begin{bmatrix}0&0&1&0\cr 0&0&1&1 …

Is there an easy way to solve the following equation for $k_1$ and $k_2$ under the constraint $|k_1|=|k_2|=1$? $\operatorname{Im}(k_1b_1 - jk_2b_2) = \operatorname{Im}(k_2b_2 - jk_1b_1) =0$ where $b …

I'm working with matrices that have large condition numbers, and I was wondering if there's a way to find a similarity transform $B = PAP^{-1}$ such that $B$ has a smaller 2-norm condition number than …

Think of it in terms of causality & superposition. $$z = f(x,y)$$ If you keep $y$ fixed then $\frac{dz}{dt} = \frac{df}{dx} * \frac{dx}{dt}$ If you keep $x$ fixed then $\frac{dz}{dt} = \frac{df}{fy …

I have a given monic polynomial $P(s)=\sum\limits_{k=0}^N a_ks^k $ of degree $N$, and I construct this matrix which has $P(s)$ for a characteristic polynomial: $$ M = \begin{bmatrix} -a_{N-1} & …

I have some measurements $M_1, M_2, M_3$ and some parameters $a,b,c$ I want to estimate. They are related as follows: $$\begin{bmatrix}M_1 \\ M_2 \\ M_3\end{bmatrix} = \begin{bmatrix}1 & -1 & 0 \\ 0 …

I'm familiar with Householder reflections; they are a simple transformation that, given a normal vector, describes reflection in the hyperplane perpendicular to that vector. But do Householder reflec …

What are the allowable transformations on the $B$ and $C$ matrices in a linear state-space system that preserve input-to-output behavior without changing the $A$ and $D$ matrices? I'm working with t …

Hmm. Here goes another attempt to muddle my way through this... For pure-diagonal modal representations with diagonal matrix $A$, the state variables are all decoupled, and therefore the entries of t …

Of course! Consider a coordinate transformation of rotation and/or scaling (but not translation): v = Au where v and u are vectors, and A is a transformation matrix. Then the eigenvectors, if they …

Sure, there are variations on least-squares approximations. Here's an engineering answer (not really a pure math answer): On one project I did at my company, we had a thermal model that was approxima …