# Tagged Questions

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### Solutions of $XA=XAX$.

All matrices are real and $n \times n$. The matrix $A$ is given. I am interested in solving $XA=XAX$. In particular, I would like some characterization of matrices that satisfy this equation. For ...
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### How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
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### Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
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### Can linearization of a function around $x=0$ show whether first derivative is positive or negative?

As title says, can linearization of a function $f(x)$ (by the method of taylor series around $x=0$) show whether first derivative of the function ($df/dx$) is positive or negative at $x=0$? And.. ...
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### Solution of a Sylvester equation?

I'd like to solve $AX -BX + XC = D$, for the matrix $X$, where all matrices have real entries and $X$ is a rectangular matrix, while $B$ and $C$ are symmetric matrices and $A$ is formed by an outer ...
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### Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...