# Tagged Questions

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### Neighbourhood of a matrix

Sometimes I find definitions which say that something happens in a neighbourhood of a matrix. For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if ...
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### Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right).$$ The first ten terms are: $0.75$ ...
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### Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many ...
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### If $P$ is an invertible transition probability matrix, does $P^{-1}[i,j]$ have any interesting meaning?

Suppose we have a Markov chain transition probability matrix $P$ that is invertible, i.e., $P^{-1}$ exists. Question: Does there exist a meaningful interpretation of the $(i,j)$ entry in $P^{-1}$? ...
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### Trying to prove a matrix is always convergent.

I have a matrix $Z$ of the form $Z = \left[Q^{-1}-Q^{-1}A^T\left(AQ^{-1}A^T\right)^{-1}AQ^{-1}\right]\Phi$ where, $\Phi$ is a diagonal matrix of real non-negative values. $\Theta$ (not ...
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### Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
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Define $$X_A(x):=A^i_{\ j}x^j$$ where $A$ is a matrix. Why is there a minus sign in the following formula? $$[X_A,X_B]=-X_{[A,B]}$$ Edit: perhaps the question is not well posed, since what I really ...
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### How to compress a linear operator and have the lossless composition property.

Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
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### Definition and some elementary properties of the “vector turn map”

This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
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### Is this map a known one?

Let $A$ be a $2\times2$ real matrix, then define $f:S^1\to S^1$ by $f(\phi)=\arctan(A\cdot(\cos \phi,\sin \phi))$. This can be viewed as a discrete dynamical system on $\mathbb{S}^1$ and I am trying ...
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### $Az=Î»z$ lead to $x(t) = c_1*e^{\lambda_1 t}z_1+c_2*e^{\lambda2 t} z_2+…+c_p*e^{\lambda_p t}z_p$ is a solution to $dx/dt=Ax$. Why?

I'm studying a course in dynamical systems. It's a pretty much linear algebra intensive course, and it's been a while since I did that sort of things. In it, they say that if vector $z$ satisfies ...
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### conditions under which real-matrix exponential are equivalent

Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$? Thanks!
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### Solution to system of difference equations with repeated unit roots

Can anyone provide the forms of the solutions for the homogeneous part and particular solutions for a non-homogeneous system of two linear autonomous difference equations ...
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### Closed form solution of this second order linear difference equation?

$$y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k$$ Transform into a system of $n$ first order equations (Step 1) \begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align} It follows that ...
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### Canonical form of a Matrix question involving a conjugacy

How do i find the canonical form of this matrix, my attempt is to use it in a conjugacy for flow. $$A=\begin{pmatrix} 0&1&0 \\ -1&0&0\\ 1&1&1\end{pmatrix}$$ do i need to ...
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### How do I know if a discrete time-invariant homogeneous dynamic system will reach, at some point, an equilibrium point?

Is this even possible? Given a time-invariant homogeneous dynamic system: $$x(k+1) = Ax(k)$$ My textbook defines an equilibrium point of the system as: A vector $\bar x$ is an equilibrium point ...
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### Flow of D.E what is the idea behind conjugacy?

I got some kinda flow issue, ya know? well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the ...
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### Changing parameters in a 3x3 Matrix trying to find the general solution.

Consider the system $$X^{'}= \begin{pmatrix} 0&0&a \\ 0&b&0\\ a&0&0 \end{pmatrix}X$$ depending on the two parameters a and b. 1) find the general solution of this system. ...
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### e'ing a matrix and then finding eigenvalues and a ivp to converge to the origion

The question is as follow's consider the Matrix A $$A = \begin{pmatrix} \lambda & 0 &0\\ 1 & \lambda&0 \\ 0&1& \lambda \end{pmatrix},$$ Compute $e^{tA}$, and use it to ...
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### Change of basis (to a more structured one) in a dynamic system, wrong result?

Given a time-invariant homogeneous dynamic system $x(k +1) = Ax(k)$, where the system matrix is: $$A = \begin{bmatrix} 1 & -1 \\ 2 & 4 \\ \end{bmatrix}$$ ...
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### Kalman filter implementation question

I have the following code to define a Kalman filter: ...
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### Quadratic equation with matricial coefficients

If I have a equation in the form $${\lambda ^2}{I_N} + \lambda {M_1} + {M_2} = {0_N}$$ where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
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### The system of $x(t+1) = Ax$ growing and retaining stability possible?

This is about general equilibrium: Suppose that $x(t)$ represents outputs of all sectors and parts of the whole economy - represented as matrix. How outputs evolve to $x(t+1)$ ...
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### logarithm of a matrix base a matrix — $\mathbf{A}^x = \mathbf{B}$

I want to solve $\mathbf{A}^x = \mathbf{B}$ where $\mathbf{A}$ and $\mathbf{B}$ are both $n$-by-$n$ matrices and $x$ is real. I see that in general there may be no solutions, or multiple solutions. I ...
I have a matrix-valued inhomogenous linear ODE $X' = F(t)X + G(t)$, $X(0) = I_{n \times n}$, $F(t),G(t) \in \mathbb{R}^{n \times n}$, and the entries of $f$ and $g$ are continuous functions. What ...