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

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0answers
19 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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0answers
25 views

On Markov triples and the square root of $2\times 2$ matrices

Let for two Markov triples $a^2+b^2+c^2=3abc,$ and $\alpha^2+\beta^2+c^2=3\alpha\beta c$, where we (can) take $a\leq b\leq c$ and $\alpha\leq \beta\leq c$. Then one has the ratio between RHS's and ...
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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
32 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
31 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 ...
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2answers
127 views
+100

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
30 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, ...
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1answer
36 views

Square Matrices and Determinant question [on hold]

For invertible matrices $A$ and $B$ of order $n$, show the following $(AB)^{-1} = B^{-1} A^{-1}$ Thank you!
0
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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}= ...
1
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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
37 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
21 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 ...
3
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2answers
62 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
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1answer
27 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 $3\times 3$ ...
0
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1answer
47 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, ...
0
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1answer
14 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
21 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
22 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 ...
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0answers
39 views

Matrix equation divide by zero

Consider a matrix equation $Ax=b$. When solving it, I decomposed $A$ with the Doolittle algorithm to its lower and upper triangular matrix, $L$ and $U$ respectively. Now, the above equation can be ...
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2answers
11 views

Orthogonal projection operator - squared of cosine

I am reading a paper making use of orthogonal projection and I came across an expression $ \frac{z^TPz}{z^Tz} $ where P = $ S(S^TS)^{-1}S^T $ which is basically the orthogonal proejction operator. z ...
2
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1answer
14 views

Is there a conceptual way to understand the formulas for the mean and covariance matrix of a conditional gaussian distribution?

Let $(x_1, x_2)^{T}$ be a multivariate normal random vector with mean $(\mu_1,\mu_2)^{T}$ and covariance matrix $$ \Sigma= \begin{pmatrix} \Sigma_{1,1}&\Sigma_{1,2}\\ ...
0
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1answer
31 views

Cubic spline interpolation results

I have a set of data points on which i am trying to do cubic spline interpolation. Below is the snapshot of the curve with the input data points marked in green color. And the red color marked point ...
4
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2answers
64 views

Prove $\exp(\mathrm{Tr}(X))=\det(\exp(X))$

Show that $\exp(\mathrm{Tr}(X))=\det(\exp(X))$ where $X$ is a matrix using the concept of the Jordan normal form I realised this formula by considering that: $\det(\exp(X))=\exp(\lambda_1) ...
1
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1answer
17 views

When does $A\mathbf{v} = \lambda B\mathbf{v}$ admit a basis of solutions?

Let $A, B \in \mathbb{C}^{n \times n}$ be Hermitian matrices, and consider the so-called generalized eigenvalue problem $$A\mathbf{v} = \lambda B\mathbf{v}$$ where $\lambda \in \mathbb{C}$ is called a ...
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0answers
15 views

Total support in matlab

I need help writing an algorithm in Matlab telling me if a radom matrix has total support or not. I'm trying to use the Linprog formula, but I don't understand it.
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2answers
53 views

Solving for an unknown symmetric matrix using an answer found by a commutator.

Suppose I have, for $A,X$ real square symmetric matrices, and $B$ skew-symmetric and real, $AX-XA=B$, with $B$ and $A$ known and $X$ unknown. What properties of $X$ need to be satisfied to find $X$ ...
1
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1answer
15 views

Translation, Scaling and Rotation of Matrix

Two 2D house models A and B are shown in the figure below. House A has one point at (3,2) and House B has one point at (0,-1). Calculate a chain of matrices that, when post-multiplied by the ...
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0answers
15 views

4x5 linear equation treaded as parameter

I got a 4x5 linear equation (4 equation 5 incognitas)like this: 1 1 0 0 0 = 800 0 1-1 1 0 = 300 0 0 0 1 1 = 500 1 0 0 0 1 = 600 i tried to give solution taking ...
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0answers
44 views

Differentitaion of a non-linear equation using FDM method

I'm a Ph.D student of Hydraulic structures. I'm reading a paper in that the equation $(II)$ below is obtained by differentiating the equation $(I)$ using FDE (Finite Difference Equation) method and ...
0
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0answers
22 views

solution of matrix equation

I was trying to solve the problem I have posted previously (here). and stuck up at the point where I need to find a simplified expression for $(\mathbf{I-DW})^{-1}$ Where $\mathbf{W}$ is a doubly ...
1
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1answer
36 views

Analytical solution to nonlinear least-squares problem

I have a data set which can be fit well to a single gaussian model, with dependent variables $y_i$ and independent variables $x_i$, with $i=1...N$. I want to avoid using a nonlinear fitting library, ...
0
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2answers
37 views

Compute the Jordan form of a particular matrix

Compute $e^S$, where $$ S = \pmatrix{ \frac 12 \ln(\alpha^2 + \beta^2) & -\arctan(\frac{\beta}{\alpha})\\ \arctan(\frac{\beta}{\alpha}) & \frac 12 \ln(\alpha^2 + \beta^2) } . $$ In ...
0
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1answer
30 views

find the set of all positive integers n for which there is a real matrix A of dimension n×n such that $A^{−1}=−A$.

Need to find the set of all positive integers $n$ for which there is a real matrix $A$ of dimension $n\times n$ such that $A^{−1}=−A$. Tried: let $\lambda$ be an eigen value of A then we have ...
0
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1answer
44 views

differentiation of a norm of matrix function

I need to differentiate the following function W.r.to $x$ $y=\|x (\mathbf{I-W}-x \mathbf{Diag(v_2)W})^{-1}\mathbf{v_1} - b\|_2$ where $0<x<\frac{2}{max_i{|{v_2}_i|}}$,$\mathbf{v_1}\in ...
0
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1answer
5 views

Proof for Inverse of the matrix

I tried to prove the inverse of the matrix but I got a wrong formula. Here's the proof: Let $$aX + bY = m \quad , \quad cX + dY = n\\ \begin{align}X & = \frac{m - by}{a}\\ & = \frac{n - ...
4
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3answers
348 views

Solve for unknown matrix

Let $A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$ and let $B = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$ Solve $A X = B$ for a matrix $X$ My guess is that i: ...
0
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1answer
39 views

What are singular value of $A$?

Let $ A = \left( {\begin{array}{*{20}{c}} {x + (\frac{3}{4} + y)i}&1&1\\ 0&{(x - \frac{5}{4}) + iy}&1\\ 0&0&{(x + \frac{3}{4}) + iy} \end{array}} \right)$, and $x,y\in ...
0
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0answers
28 views

Derivative of $\frac{dA^TA}{dx}$ [duplicate]

I think I have an easy question but I can't understand how to do. I am trying to figure out how to do the derivative of a symmetric matrix $B = A^TA$ w.r.t. one parameter $x$ of the matrix: ...
1
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0answers
34 views

Product of matrix-valued normal densities and Kronecker product

I am trying to find an expression for the mean, column-covariance and row-covariance matrices of the product of two matrix-valued Normal distributions. Here is what I've tried in a special case I ...
0
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1answer
15 views

Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations. Let $A$ be an ...
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2answers
34 views

An example of unitary matrix which is $3\times 3$ and complex

Please give me an example of unitary matrix which is $3\times 3$ and complex. If I get this example, i will finish my thesis.
1
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0answers
22 views

Solving a matrix equation problem

I've been trying to solve this matrix equation but I just can't find the way to do it. The task is: Find matrix $\mathbf{X}$ from the equation in the photo where $\mathbf{A, B, C}$ are given matrices ...
3
votes
1answer
69 views

Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
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0answers
18 views

Question with Gauss-Jordan elimination produces the matrix

Please help me explain the problem below: How can I use Gauss-Jordan to get all bottom roll become 0? Thank you so much!
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2answers
25 views

solving an XOR matrix

I'm working on a somewhat-unique linear algebra problem arising from XORing files together in order to encode them, and then figuring out how to subsequently recreate the original files from the ...
0
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1answer
18 views

Consistency of system of linear equations without taking it to echelon form

Establish the conditions under which the equations $$ax + by + cz = q-r;bx + cy + az = r-p;cx + ay + bz = p-q ,$$ are consistent. I am aware that by taking the system to echelon can get me the rank ...
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0answers
14 views

Commutivity with Jordan Decomposition of Linear Transformation

Let $T \in L(V,V) $ and $T = D + N$ Where $D + N$ is the jordan decomposition of $T$ and $D$ is diagonable Let $X \in L(V,V)$ Show $XT = TX$ iff $XD = DX$ and $XN = NX$ I started out with saying ...
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1answer
19 views

Why some of the value in inverse matrix become positive?

Ans: But why? Isn't it calculate something like this: Please help explain. Thank you!
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1answer
21 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...