Questions related to equations, with matrices as coefficients and unknowns.

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0answers
16 views

Solving a generalized eigenvalue problem with constraints

I have the following generalized eigenvalue problem: $ \begin{pmatrix} 0 & a \\ a^T & B / \lambda_{i} \end{pmatrix} \begin{pmatrix} 1 / \lambda_{i} \\ x_{i} \end{pmatrix} =\epsilon_i \begin{...
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4answers
42 views

If $Y = X\beta$ are a system of linear equations and that $X$ is NOT full rank. Is this system under or over determined?

Suppose I have a system of linear equations, $Y = X\beta$, where $Y$ is a $n$ by $1$ matrix, $X$ an $n$ by $n$ matrix, and $\beta$ a $n$ by $1$ matrix. Suppose that I know what $Y$ and $X$ are, and ...
1
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3answers
30 views

Solve for $X$ in matrix equation

How can I solve for $X$ in this matrix equation? $$\begin{bmatrix}-3&-8\\-9&5\end{bmatrix} X + \begin{bmatrix}4&-7\\3&-2\end{bmatrix} = \begin{bmatrix}5&8\\-1&-1\end{bmatrix} ...
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0answers
10 views

Convergence of a stationary iteration method for linear systems

Recently, I obtain a linear system, $Ax = b$, where $A$ is a nonsingular, strictly diagonally dominant $M$-matrix. Then I also got a matrix splitting $A = S - T$, where $S$ is also a nonsingular, ...
1
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2answers
72 views

Solving this matrix equation.

Given the following matrix equation, $$\begin{bmatrix}a && b \\ c&& d\end{bmatrix}^n\begin{bmatrix}\alpha\\\beta\end{bmatrix}=\begin{bmatrix}\gamma \\ \kappa\end{bmatrix}$$ $\alpha, \...
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1answer
53 views

Matrix addition and eigen values/vectors

If I start with matrix A given by $A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}$ and I express it as a sum $A = \begin{bmatrix} w & x \\ y & z ...
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0answers
93 views
+50

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
0
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0answers
25 views

How to represent the summation in matrix equation

I have two vectors $a=[a_1, a_2,..a_n]$ and $b=[b_1,b_2,..b_n]$. A vector $c=[c_1...c_n]$ where $$c_i=\sum_{j=m}^l\alpha_j a_j+\sum_{j=k}^ h\beta_j b_j$$ In which, $i=1, \cdots, n; m,k \ge 1, m \le l ...
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1answer
59 views

Can someone come up with a better way to write $V = \operatorname{diag}(x_1,x_2)(Y-\mathbf{1}X^TY)$

$\newcommand{\diag}{\operatorname{diag}}$Let $X = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $Y= \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$ I have a vector: $$V = \begin{bmatrix} x_1(y_1 - \...
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0answers
43 views

Solving The system subject to the given initial conditions

My question relates to question b. In this question, I have made the differential equations into a Matrix and found the eigenvalues to be 5 (with multiplicity of two). However, the dimension of the ...
1
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1answer
52 views

How to solve this quadratic matrix equation?

I have to solve the following equation where D is the unknown matrix: $$D^{T}D(DVD^{T}+I)^{-1}=A$$ I is the identity matrix, V and A are know constant matrices. Does anyone have any idea how to solve ...
3
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1answer
47 views

What is the name of this problem? linear Matrix equation optimization?!

I have almost no knowledge in linear algebra but I need to understand the process of solving a problem. In fact I'm looking for some keywords or hints to know what exactly should I be Googling! So any ...
0
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2answers
76 views

If $X$ is zero matrix,what is $e^X$?

Let $X$ be an n×n real or complex matrix. The exponential of $X$, denoted by $e^X$, is the n×n matrix given by the power series $e^X =\sum_{k=0}^{\infty} X^k/k!$ where $X^{0}$ is defined to be the ...
0
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1answer
15 views

Significance of a capital R outside of brackets containing a matrix expression

I was trying to understand what a positive definite matrix is while reading a reinforcement learning paper today, and I came across this page: http://mathworld.wolfram.com/PositiveDefiniteMatrix.html ...
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5answers
130 views

Calculating the matrix $M^{2006}$

Say you have the matrix $M$: $$\begin{bmatrix}\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{bmatrix}$$ How do you find $M^{2006}$? My thinking was that you ...
0
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2answers
33 views

Solve the following systems of

Solve the following systems of equations by matrix method $$2ax-2by=-a-4b$$ and $$2bx+2ay=4a-b$$ I only need the equation in terms of $x$ and $y$ in order to represent in the matrix form but how can ...
0
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1answer
35 views

Given the parallel and perpendicular component of a vector in terms of another vector, how do you determine the tensor that connects both?

Sorry for the awkwardly phrased title, I wasn't sure how to properly word it. I want to do the following: I have a vector $\vec J$ and a vector $\vec E$ with the following relation (with the ...
0
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0answers
15 views

Skew symmetric and Telescopic operator

I have been reading some papers that refer to Skew-symmetric and telescopic operators. For eg. this is a link to a presentation showing the concept. Presentation Now, I understand what one means by ...
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3answers
64 views

Solving a third degree matrix equation

Let $X, A, B$ be square, matrices, and let $X$ be an invertible covariance matrix (symmetric, square, positive definite). Is it possible to solve for $X$ the following equation? $$ A=X(I+B-XX) $$ ...
2
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2answers
62 views

General matrix solution for a variant orthogonality condition

An $n\times n$ complex matrix $X$ satisfies a constraint looking like a rank deficient version of the orthogonality condition $$ X^TX = \text{diag}\left(1,\dots,1,0\right), $$ where $X^T$ is the ...
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2answers
40 views

How to solve this matrix equation

Consider the system of ODE in $\Bbb R^2 $ $\dfrac{dY}{dt}=AY$ where $Y(0)=$ \begin{bmatrix} 0 \\ 1\end{bmatrix} $t>0$ where $ A=$ \begin{bmatrix} -1 & 1 \\ 0 & -1\end{bmatrix} and $Y(t)...
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2answers
40 views

How to Solve a linear matrix equation of an array $M = BMC$ where $ B$ and $C$ are known

Adding to the question's description : I am doing Feature extraction from videos and i am trying to implement this one line of mathematical equation to matlab or even any algorithm . let's say I ...
1
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1answer
30 views

Matrix for zeroing specific entries.

Is there a matrix, $X$, that can be solved-for here? $ \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right) X = \left( \begin{array}{ccc} 0 &...
1
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1answer
28 views

Reconstructing a matrix

Before reading on, let me acknowledge that this problem is solveable generally, however I am interested in knowing if a certain form of solution exists. If I have a square complex unitary $n\times n$...
0
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1answer
32 views

Matrix Optimization of $\max_{A} \mathrm{Trc} \left((I-A \cdot B) \cdot M \cdot (I-A \cdot B)^T + A KA^T \right)$

Suppose we have to positive definite matrices $M$ and $K$. I want to optimize the following expression \begin{align} \max_{A} \mathrm{Trc} \left((I-A \cdot B) \cdot M \cdot (I-A \cdot B)^T + A KA^T \...
1
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1answer
34 views

Conditions for uniqueness of solution to a linear system of equation

Consider a $n\times n$ M-Matrix $\mathbf{A}$ and a $n\times n$ non-negative and non-zero matrix $\mathbf{B}$. Also, let $\mathbf{x}$ and $\mathbf{b}$ be two (non-zero) n-column vectors. I am looking ...
0
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2answers
27 views

Find all values of $k$ for which the given augmented matrix corresponds to a consistent system

Consider the augmented matrix \begin{bmatrix}1&k&-1\\4&8&-4\end{bmatrix} I want to find all the values of $k$ for which the corresponding linear system is consistent. Upon dividing ...
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0answers
16 views

Solution of two inter related real symmetric system

Let P and Q be two N by N real symmetric matrices. X and Y are two N by 1 dimensional vectors with real elements . The following equations are satisfied: PX=QY PY=QX P and Q can be written as: P=A+...
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1answer
32 views

Prove $(A^{T}+I)(A^{T}-I) = (A^{T}-I)(A^{T}+I)$ [closed]

Here $A^{T}$ is transpose matrix of $A$, $I$ is identity matrix. Also prove 1) $(I-A)(I+A) = (I+A)(I-A)$ 2) $(I-iA)(I+iA) = (I+iA)(I-iA)$ 3) $(I-iA)^{-1} (I+iA)^{-1} = (I+iA)^{-1}(I-iA)^{-1}$ ...
0
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1answer
47 views

Prove this matrix algebraic equation

Let $X$ be an $n \times m$ matrix, $x_i^T$ the $i$th row of $X$. Is it true that $\sum_{i=1}^n x_i^T (X^T X)^{-1} x_i = m$ ? How do I prove it?
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0answers
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Lanchester's Square Law – Complex armies

Lanchester's Square Law states that given two armies, $x$ and $y$, with the army units' relative strengths $\alpha$ and $\beta$, respectively, we can write two differential equations for the sizes of ...
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2answers
33 views

number of solutions of system of two equations, two unknowns (Matrix)

How can we find that when a system of two equations, two unknowns has Infinite Solutions. I want a solution with matrix. I know this method (which is not with matrix): $ax + by = c$ $a'x+ b'y = c'$ ...
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0answers
23 views

Equations involving orthogonal matrices

My variables are: $O_1,\dots,O_M\in \mathbb{O}(d)$ I have equations like: $O_{1_1} x_{1,1}+ \dots + O_{1_{m_1}} x_{1,m_1} = x_{1,(m_1+1)}$ $\vdots$ $O_{n_1} x_{n,1}+ \dots + O_{n_{m_n}} x_{n,m_n} =...
2
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0answers
44 views

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 ...
0
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1answer
8 views

Proof of non-singularity given certain conditions

Suppose that I have a $n\times t$ matrix $\boldsymbol{X}$ that is full rank and a non-singular matrix $\boldsymbol{L} = \begin{bmatrix} \boldsymbol{L}_1 & \boldsymbol{L}_2 \end{bmatrix}$ such that ...
2
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0answers
36 views

Eigenvectors of matrix equation $AX+XB^\text{T}=\lambda (CX+XD^\text{T})$

We have the matrix eigenvalue problem $$AX+XB^\text{T}=\lambda (CX+XD^\text{T})$$ Where $\lambda$ is the eigenvalue, $X$ is $m\times n$ and plays the role of the eigenvector, and $A$, $B$, $C$, and $...
3
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0answers
36 views

Eigenvectors of Generalized Sylvester Equation $AX+XB^\text{T}=\lambda CXD^\text{T}$

Ok here's what I mean with the Sylvester equation eigenvectors. The simplest case, where $C = D = I$, has already been solved in the literature (Matrix Calculus by W.H. Steeb). $$A X + X B^\text{T}=\...
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2answers
228 views

Given a symmetric positive-definite matrix $M$, find all $A$ such that $A^\top M A=M$

Given $M$ a real symmetric positive-definite matrix, I would like to characterise all matrices $A$ such that $A^\top M A=M$. Note that the question of finding $A$ solutions to $A^\top M A=M$ for all ...
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1answer
34 views

Find vector x such that matrix multiplication Sx = 0

I have the following matrix $$S= \begin{bmatrix} -1 & 1 & 0\\ -1 & 1 & 1\\ 1 &-1& -1\\ 0 &0 & 1 \end{bmatrix} $$ I wish to find a non-negative, non-...
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0answers
20 views

Linear algebra: Solving for the coefficients on vectors

I am solving the following system: $$ -\frac{1}{r^2}\begin{bmatrix}\sqrt{\mu}\cos(\theta)\\ \sin(\theta) \end{bmatrix}= (v_r'-v_{\theta}\theta')\begin{bmatrix}\frac{\cos(\theta)}{\sqrt{\mu}...
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3answers
45 views

Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
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1answer
40 views

Finding all orthogonal matrices commuting with a positive-definite matrix

Given $M$ a symmetric positive-definite matrix, I'd like to characterise the orthogonal matrices $Q$ commuting with $M$: $MQ=QM$. $Q$ and $M$ commute if and only if they are simultaneously ...
2
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1answer
28 views

Using inverse of transpose matrix to cancel out terms?

I am trying to solve the matrix equation $A = B^TC$ for $C$, where $A$, $B$, and $C$ are all non-square matrices. I know that I need to utilize $M^TM$ in order to take the inverse. I'm just not sure ...
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2answers
74 views

Solution of $A^\top M A=M$ for all $M$ positive-definite

I am trying to find all matrices $A$ such that for all positive-definite matrices $M$, $A^\top M A=M$. $I$ and $-I$ are obvious solutions. I can't find out it there are other such matrices and if so, ...
3
votes
1answer
28 views

Shapes described by a homogeneous quadratic equation

Suppose we have a homogeneous quadratic equation of three variables $w_1$, $w_2$, and $w_3 \in \mathbb{R}$ as follows: $$W^TAW=0.$$ where $W=[w_1,w_2,w_3]^T$ and $A$ is a non-singular $3\times 3$ ...
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1answer
48 views

How do I rearrange this matrix equation to find A and b?

The Question: It is possible to rearrange the matrix equation $\pi^TP= \pi^T$ into a linear system $Ax = b$ where $x = \pi$ is the unique solution to the system. Such a system could be solved by, ...
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1answer
15 views

How can I find the values of each coefficient within a vector in a matrix multiplication problem?

I have all the other T and alpha values and I'm trying to solve for a0 a1 and a2 I can't simply divide them so I figure in need to do something like this But i'm not sure how to go about it? ...
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0answers
27 views

Matrix multiplication and correct brackets placing

I have sequences of brackets like this [ > ) ]. I have to add brackets to the sequence that result would appear in this way ...
2
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0answers
39 views

Generators of a matrix group

I have just read that the group $\Gamma_1(6)=\left\{ \begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z}):a\equiv d\equiv 1 \mod 6,\, c\equiv 0\mod 6\right\}$ is generated by the matrices $...
1
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2answers
24 views

How to shear a matrix so that the parallelogram formed by its vectors has right angles.

In my lecture today we were told that the area of a parallelogram with sides given by the vectors $v_{1} \,\,v_{2} \in \mathbb{R}^{2}$ is equal to the absolute value of the the determinant on the ...