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

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1
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2answers
41 views

Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
0
votes
0answers
24 views

Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
0
votes
1answer
28 views

how to solve an matrix equation that is similar to a sylvester equation

during an algorithmn, I have to solve an equation of the form $$AXD-XBD=C$$ with $A\in\mathbb{R}^{n\times n}$,$X\in\mathbb{R}^{n\times m}$,$B\in\mathbb{R}^{m\times m}$,$D\in\mathbb{R}^{m\times p}$ and ...
2
votes
2answers
57 views

What are the constraints on $\alpha$ so that $AX=B$ has a solution?

I found the following problem and I'm a little confused. Consider $$A= \left( \begin{array}{ccc} 3 & 2 & -1 & 5 \\ 1 & -1 & 2 & 2\\ 0 & 5 & 7 & \alpha \end{...
0
votes
1answer
39 views

Intuitive way to understand the use of matrix inversion to find dual basis

I'm currently thinking about the following problem: Problem: Let $B = (b_1, b_2, b_3)$ a base of $\mathbb{R}^3$. Find the correlating dual basis $B^* = (b_1^*, b_2^*, b_3^*)$. $B$ is explicitly ...
1
vote
1answer
23 views

Why is there a division by $z$ in this mathematical model of a camera?

I'm reading the OpenCV documentation on what mathematical model they use for a camera. The below quoted text can be found on this website, scrolling down to the section "Detailed Description". I do ...
0
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0answers
25 views

Concentrated Likelihood

I am working on Panel Data models and I am having some issues to obtain the concentrated log-likelihood function for the following model. $$ y_{it} = \delta'x_{it} + \beta_i'f_t + u_{it} , i = 1,...,N,...
0
votes
1answer
9 views

projectors solution of a matrix equation

If I have two n*n dimensional complex matrix A and B, A and B are both projectors. i.e., A^2=A and B^2=B. If A*B=A and A is not identity, clearly if B=A, or B=I then the equality holds, can B has ...
5
votes
1answer
32 views

a property of infinite matrices

An infinite matrix $[a_{ij}]_{i,j\in\mathbb{N}}$ is called invertible, if for any convergent sequence $(y_m)$ there exists exactly one sequence $(x_m)$ such that $y_m=\sum_{n\ge 1}a_{mn}x_n$ for all $...
-2
votes
1answer
40 views

Linear Algebra Eigenvalues and Eigenvectors [on hold]

So I have a 2x2 matrix where equation 1(EQN1) is 1 and 2; equation2(EQN2) 2: 4 and 3 The determinant is det(A-λI)=0 When I first solve the eigenvalues I get λ=5, λ=-1 Now this is where I am lost,...
0
votes
1answer
24 views

How to find a scalar given a matrix equation with an unknown matrix?

I am not expert in linear algebra. I couldn't come up with any solution for my problem with my limited knowledge. So the question may be even silly or have no solution, I don't know. But I appreciate ...
0
votes
0answers
16 views

How can I take a distance matrix and construct a coordinate representation from it

Say I have a distance matrix M of rank n, where the distance between the ith and jth point is M[i,j]. the diagonal of such a matrix will be 0. How can I convert the distance matrix M to coordinates ...
1
vote
1answer
31 views

Condition for Linear Dependence

Let $\mathbf{x}\neq \mathbf{0}$ and $\mathbf{y}\neq \mathbf{0}$ be $n \times 1$ vectors, $\mathbf{A}\neq \mathbf{0}$ and $\mathbf{B}\neq \mathbf{0}$ be $m \times n$ matrices with $m>n$, and, for ...
0
votes
2answers
55 views

How can I find $E_1$ and $E_1^{-1}$?

Suppose that: $E_1 \begin{bmatrix}12\\35\end{bmatrix} = \begin{bmatrix}48\\35\end{bmatrix}$ Find $E_1$ and $E_1^{-1}$ I know how to find the inverse of a matrix, that's easy. It's just $E_1^{-1} = ...
0
votes
1answer
37 views

Number of Rational Solutions $\mathbf{x}\in[0,1)^n$ to the Matrix Condition $\mathbf{A}\,\mathbf{x}\in\mathbb{Z}^n$

Let $n$ be a positive integer and $\mathbf{A}$ an $n$-by-$n$ matrix with integer entries. Suppose that $k:=\big|\det(\mathbf{A})\big|$ is nonzero. How many $n$-by-$1$ column vectors $\mathbf{x}\in\...
0
votes
1answer
22 views

Find conditions on $x$ and $y$ which guarantee that one can locally solve the following for $u(x, y)$ and $v(x, y)$

My understanding of this question is that I need to show that the following equations can be solved where $u$ and $v$ can be written as a function of $x$ and $y$. $xu^2+yv^2=9$ $xv^2-yu^2=7$ I ...
0
votes
0answers
16 views

How to proof $M(H,K)X=U([M(\lambda_{i},\mu_{j})]_{ij}\circ (U^{*}XV))V^{*}$

Let $M(x,y)$ be positive real function on $(0,\infty)\times (0,\infty)$ satisfies $M(x,y)=M(y,x)$ $M(\alpha x,\alpha y)=\alpha M(x,y)$ for all $\alpha>0$ $M(x,y)$ is non-decreasing in $x$ and $y$...
0
votes
1answer
27 views

sign the elements of a ugly matrix

I have a matrix defined as $A=(I-aT)^{-1}F$, where I is identity matrix, a is a positive constant smaller than 1, T is a stochastic (transition matrix) and F is a matrix with positive diagonal ...
0
votes
1answer
34 views

Linearly Dependent Equations

Let $\mathbf{x}$ and $\mathbf{y}$ be $n \times 1$ vectors, $\mathbf{A}\neq \mathbf{0}$ and $\mathbf{B}\neq \mathbf{0}$ be $m \times n$ matrices with $m>n$, and, for some $u < m$, let $$ \...
2
votes
0answers
22 views

Is Singular value decomposition suitable for solving matrix equations Ax=b?

Consider the following equation $A_{n\times n}x_{n\times 1}=b_{n\times 1}$. This is a nonhomogeneous linear system, but the SVD is "specialized" for solving homogeneous systems, like $Ax=0$. This ...
1
vote
1answer
51 views

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix?

How to obtain the inverse of $MSM^T$ when $(MM^T)^{-1}$ is already known and $S$ is an invertible symmetric matrix? Assume that $M$ is an $n \times m$ matrix with $n \leq m$. Is it possible to obtain ...
0
votes
0answers
22 views

How to solve incomplete system of linear equations

I need to analyze a significant data set manually, and I'm looking for some math love to try to bring interesting values on top so that my work is easier. I'm trying to devise a sorting criteria for ...
1
vote
0answers
30 views

Conditions for a block matrix to be positive definite

Consider the block matrix given by $$M = \begin{bmatrix} A_{11}&A_{12}&0\\ A_{12}&A_{22}&A_{23}\\ 0&A_{23}&A_{33}\end{bmatrix}$$ What conditions should I impose on each ...
0
votes
2answers
38 views

Sylvester equation

Solving the Sylvester equation $A*X+X*B = C$, where $A$ and $B$ are similar. I am aware that the solution is non-unique, however I have the information that all entries in $X$ are positive. With ...
2
votes
3answers
44 views

If $BNA=N$ it can happen that $B$ and $A$ identities?

Consider two matrices $A\in GL_m(K)$, $B\in GL_n(K)$ such that for any $N\in M_{n,m}(K)$ is true that $$BNA=N$$ How can I prove that $B$ and $A$ are identities?
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votes
1answer
28 views

Matrix summation

$ \left( \begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array} \right) % \left( \begin{array}{cc} a \\ b \end{array} \right) % = % \left( \begin{array}{cc} 3 - X \\ 6 - X \end{array} \right) $ Can ...
-1
votes
0answers
11 views

Algorithm to find least distant locations in a distance matrix?

Yellow color are the location codes Green square is the matrix with places that are closely located. The other possible closest places are grey shaded. If someone has liberty to choose number of ...
2
votes
1answer
46 views

Construct a sequence of matrices

Let $\varepsilon>0$. Suppose $A,B\in M_n$ and $\det(A+B)=0$ and ${\left\| B \right\|_2} \le \varepsilon $. I need construct a sequence of matrices $\left( {\left\{ {{\Delta _i}} \right\}_{i = 1}^...
0
votes
0answers
108 views

Calculating probabilities in a Markov chain process [on hold]

I have 3 variables A, B and C with each variable having a probability of 0.6 and 0.4 i.e. A can have states (ON) with probability of 0.6 as well as can remain in certain states (OFF) with probability ...
1
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0answers
43 views

Given matrices A and B, how does one find C such that $ C^{-1}AC = B$?

Given matrices A and B, how does one find C such that $ C^{-1}AC = B$ ?
2
votes
1answer
16 views

Fractional powers of affine matrices

Take a rubber gasket. Make a slice from the middle to the outside, like the first cut in a pie. Because there was some strain in the rubber, the gasket doesn't close into a ring now, it's more like ...
0
votes
1answer
16 views

Complexity of LUP decomposition of tri-diagonal matrix to solve an equation?

Doing LU decomposition of tri-diagonal matrix and then solving the eqn by using forward substitution followed by backward substitution is done is O(n) time. http://www.cfm.brown.edu/people/gk/chap6/...
8
votes
4answers
119 views

Matrix equation $A^2+A=I$ when $\det(A) = 1$

I have to solve the following problem: find the matrix $A \in M_{n \times n}(\mathbb{R})$ such that: $$A^2+A=I$$ and $\det(A)=1$. How many of these matrices can be found when $n$ is given? Thanks in ...
3
votes
1answer
48 views

How to solve the following quadratic matrix equation?

Solve the following matrix equation in $D$ $$ A=D^{T}(DVD^{T}+\alpha\lambda_{\max}(D^{T}D)I)^{-1}D$$ where $I$ is the identity matrix, $A$ and $V$ are known matrices, $\alpha$ is a known ...
0
votes
1answer
35 views

Equivalent notations for second partial derivative of a quadratic form

I noticed this notation while going through a tutorial on matrix calculus: $$ \frac{\partial x^TAx}{\partial xx^T} = \frac{\partial}{\partial x}\left( \frac{\partial x^TAx}{\partial x} \right) = A^...
0
votes
1answer
29 views

How to solve AX=B for some given 3x3-matrix A and 3-vector B [closed]

I want to solve the following equation $$AX=\left( \begin{matrix} 9 \\ 3 \\ -3 \\ \end{matrix} \right) $$ where $$A=\left( \begin{matrix} 1 & -2 & -2 \\ -2 & 1 & -...
1
vote
2answers
55 views

How to find largest coefficient in matrix?

$$\begin{bmatrix} 1&2&6\\ 7&8&3\\ 0&4&7 \end{bmatrix} $$ I want know the algorithm to find largest value in matrix .
1
vote
1answer
59 views

Can someone suggest a way to simplify $x_1(y_1 - x^Ty) + x_1(w_1 - x^Tw)^2 - x_1^2(w_1 - x^Tw)^2 + x_1x_2(w_1 - x^Tw)^2$

Let $x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$, $y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}$, $w =\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$ I have the following vector: $V = \begin{bmatrix} ...
0
votes
0answers
24 views

Matrix Algebra equation under a constraint

I have toiled hard with this problem but I have neither been able to find a solution, not prove that no solution exists. $W_1 \in \mathbb{R}^{m \times k}, W_2 \in \mathbb{R}^{n \times k}, X \in \...
0
votes
4answers
49 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
vote
3answers
33 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} ...
1
vote
0answers
12 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
vote
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, \...
0
votes
1answer
60 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 ...
7
votes
0answers
123 views

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
votes
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 ...
1
vote
1answer
60 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 - \...
1
vote
1answer
62 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
votes
1answer
48 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
votes
2answers
77 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 ...