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2 votes

Finding value of unknown coefficient such that a linear system has a number of solutions

Here is a bit of an unconventional approach, but I think it works nicely here: Take the first two equations and transpose them: $2x+y=1-3z$ $(1)$ $x+2y=2-2z$ $(2)$ Now, solve these for $x$ and $y$ in ...
Red Five's user avatar
  • 2,762
0 votes

How to solve abstract matrix equations involving transposes and inverses?

These are pretty much the main things to keep in mind: Matrix addition works exactly the same as you'd expect. You can add or subtract matrices from both sides of an equation. Matrix multiplication ...
Wyatt Kuehster's user avatar
1 vote

What is the square root of a square matrix squared?

You already can't conclude this for ordinary real or complex numbers; the best you could possibly ask for is that $A = \pm x$ (by which I mean $A = \pm x$ times the identity matrix). But actually you ...
Qiaochu Yuan's user avatar
0 votes

A general solution $M$ for the matrix equation $A^* M^* + M A = 0$

This is indeed a Sylvester-like equation, but the adjoint on $M$ prevents us to solve it in the same general way $-$ apart from particular solutions such as $M = iA^\dagger$ or by assuming that $M$ is ...
Abezhiko's user avatar
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2 votes

Given $A_i, B_i \in \mathbb{R}^{k \times d}$, minimize $\sum_{i} \lVert U A_i V^T - B_i \rVert_F^2 $ over orthogonal $U, V$.

$ \def\BR#1{\Big[#1\Big]} \def\LR#1{\left(#1\right)} \def\op#1{\operatorname{#1}} \def\trace#1{\op{Tr}\LR{#1}} \def\frob#1{\left\| #1 \right\|_F} \def\q{\quad} \def\qq{\qquad} \def\qif{\q\iff\q} \...
greg's user avatar
  • 36.8k
0 votes

Finding the derivative of trace $AXBXC^T$ with Respect to $X$

By invariance of the trace under rotation of the factors and cancelling of any basis change matrices inside, in Einstein notation in any basis $$d (A_{ij} X_{jk}B_{kl}X_{lm}C_{im})= A_{ij} dX_{jk}B_{...
Roland F's user avatar
  • 3,031
0 votes
Accepted

Finding the derivative of trace $AXBXC^T$ with Respect to $X$

You might want to employ the index notation and make use of the definition of the trace and partial derivative. Using Einstein summation convention - implied summation over repeated indices - you may ...
Egor Larionov's user avatar
5 votes

If $A$ is a matrix such $( A-2I ) ( A-4I ) = 0$, why can't I assume $A = 2I$ or $4I$?

Have a look at what happens for $$A=\begin{pmatrix} 2 & 0\\ 0 & 4 \end{pmatrix}.$$ The real matrices ring has zero divisors when $n \ge 2$. Here, $A-2 I$ and $A -4I$ are singular matrices ...
mathcounterexamples.net's user avatar
6 votes

If $A$ is a matrix such $( A-2I ) ( A-4I ) = 0$, why can't I assume $A = 2I$ or $4I$?

Matrices are not an integral domain, i.e. the product of two non-zero matrices can be null. For instance with $$A = \begin{pmatrix}0&0\\ 1&0 \end{pmatrix} \quad B = \begin{pmatrix} 1&0 \\0&...
Florian Ingels's user avatar
0 votes

Solving a system of coupled differential equations using eigenvalues and eigenvectors

The eigenvalues of your matrix are $\lambda=-g\pm\sqrt{g^2-\Delta^2}$. Hence, the system of ODEs in the new set of coordinates $(\{y,z\}\to\{\tilde y,\tilde z\})$ becomes $$\dfrac{\mathrm d}{\mathrm ...
Conreu's user avatar
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