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

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0
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1answer
10 views

Solution to tensor/matrix equation

I need to find a real, symmetric matrix, $A$, that satisfies: $\sum_{i,j} c_i c_j A_{ki}A_{jl} = A_{lk}$ I believe this is an equation of the form: $c^T B c = A$, where $c$ is $\mathbb{R}^{N \times ...
0
votes
1answer
36 views

Find function by 2 tangents and 2 points

I am looking for explicit function descriptions $F_1(s)$ and $F_2(s)$, following the line plotted. The line is just a description, but $F_1$ should never exceed $F_m$ and start at $s_0$ with a tangent ...
-1
votes
1answer
21 views

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $? [on hold]

Let $ \alpha \neq 0 $ isn't $ n \times n $ identity matrix and $ P $ is $ n \times m $ matrix. Let $ P = \alpha P $ . When $ P \neq 0 $ ?
2
votes
1answer
31 views

How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
2
votes
4answers
145 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
0
votes
1answer
13 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
1
vote
0answers
36 views

how to rearrange matrix equation to have unknown in vector form

I am looking for the name/type of following equations: $$\dot{\theta}\dot{J} = \ddot{x} - J\ddot{\theta}$$ here the unknown is $J \in R^{m \times n}$, $x \in R^{m \times 1}$, $\theta \in R^{n \times ...
0
votes
2answers
49 views

Condition for vector to have redundant coordinates

Consider the vector $V\in\mathbb{R}^6$: $$V=\begin{bmatrix} -1&1&-1&-1 \\ -1&1&1&-1 \\ -1&-1&1&-1 \\ -1&1&1&1 \\ -1&-1&1&1 \\ ...
3
votes
1answer
38 views

How is the orthogonal projection on to the span of the columns of a matrix determined by a chosen inner product?

I know that of course a orthogonal projection must be orthogonal for a chosen inner product. But how can I find a new orthogonal projection based on $P=A(A^TA)^{-1}A^T$, if I have dot product defined ...
0
votes
1answer
24 views

Common solutions to quadratic equations associated to self-adjoint matrices

Let $\mathcal{H}$ be a complex Hilbert space of dimension $d<+\infty$, and let $\{|n\rangle\}$ with $n=0,\cdots,d$ be an orthonormal basis in $\mathcal{H}$. Let $\mathbf{A}$ be a self-adjoint ...
0
votes
1answer
24 views

how to solve a matrix equation like that

$$M^{T}MY + \lambda Y = D$$ M, Y, D are matrix, $\lambda$ is scalar value, Y and D have the same dimension. M and D is known, how to solve Y.
2
votes
1answer
36 views

Minimizing variance subject to linear inequality

Let A be a $n \times n$ matrix. Where $A$ is a symmetric positive definite matrix. Let $b$ be a vector in $R^n$. $x$ is an unknown vector to be determined. I'm interested to find vector $x$ such ...
0
votes
2answers
38 views

Is it possible to get the original eigenvector after scaling a matrix?

Let ${\mathbf{X}}\in\mathbb{R}^{n\times 1}$ and ${\mathbf{Y}}\in\mathbb{R}^{n\times 1}$ and let $\mathbf{A}\in\mathbb{R}^{n\times 2}$ be defined as \begin{equation} \mathbf{A} = ...
1
vote
0answers
24 views

Linear matrix transformation that keeps the first nonzero element in each row of a matrix and zeros the rest

I need to find a matrix transformation that takes a (non-square) matrix and within each of its rows keeps the first nonzero element in that row and zeros out the rest of the entries within that row. ...
0
votes
0answers
52 views

How to reach this result using only matrix operations?

$$\mathbf{Xw}-\lambda_w \mathbf{1}=0$$ Where $\lambda_w$ is a scalar, $\mathbf{w}$ is a vector, and $\mathbf{X}$ is a symmetric p.d. matrix. It is known that $\mathbf{1'w}=1$ (but I don't think that ...
-1
votes
1answer
30 views

Determine all values b1, b2 such that the following system has no solution

Determine all values $b_1$, $b_2$ such that $x_1+2x_2-x_3=b_1$ $-2x_1-4x_2+2x_3=b_2$ $x_1-x_2+x_3=2$ has no solution. I know i want to do row echelon form after i put the system into a matrix i ...
0
votes
0answers
13 views

What do we know about rank-2 perturbations?

Are there any theorems known about the changes in spectrum of a matrix A when it is changed to A+X, when X is rank-2? I am particularly interested in the case when X is a zero matrix except for ...
0
votes
1answer
28 views

Matrix equation from optimization problem

I am having a problem to find the solution to the following equation which has arisen as part of the solution of a (convex) optimization problem I am considering. $$\left(\frac{a}{n ...
0
votes
1answer
16 views

Is it possible to solve for scalar in this multiplication of two quadratic forms involving inverse matrix?

Given the following two quadratic forms: $$a^2=\mathbf{w'Xw}$$ $$b^2=\mathbf{1'X^{-1}1}$$ And the known relations: $$a^2b^2=1$$ $$\mathbf{X}=\mathbf{\Sigma}-\lambda\mathbf{R}$$ Where ...
0
votes
0answers
23 views

$AX=B$: How to solve for $X$ if elements of matrix A are matrices

Objective: I am trying to solve for $C$ in 2D space (x,y) and time from following PDE. $$\begin{align} \text{PDE: }\frac{\partial C}{\partial t} + \nabla\left(v.C - D\nabla{C} \right)= \alpha.C ...
4
votes
3answers
160 views

problem of a positive definite matrix

Let $H$ be a positive definite matrix and $I$ the identity matrix. If $k_1,k_2>0$, can we conclude that $k_1H-k_2I$ is positive definite if $k_1\gg k_2$?
2
votes
2answers
39 views

Using Least Squares to calculate a matrix in an equation.

I have two sets of vectors $v_i$ and $w_i$, in some $d$ dimensional space. I know that $v_i \approx M \cdot w_i$ for all i. I.e., I know that the $v$ vectors are a linear transformation of the $w$ ...
0
votes
0answers
22 views

How to solve for a scalar factor in a matrix equation?

According to the paper Motion and structure from line correspondences; closed-form solution, uniqueness, and optimization (Weng et al.), Equations (2.20 cont) there is a Matrix Equation $$ ...
1
vote
0answers
27 views

trace calculation of an operator valued matrix

Heyho, i've got problems understanding a certain calculation of the trace of an operator valued matrix right now. We've got the Matrix $T(\lambda)= \begin{pmatrix} A(\lambda) && B(\lambda) ...
0
votes
1answer
45 views

Prove that the given block matrix is positive semi-definite

How do I show $M = \begin{bmatrix} A & B \\ B^T & C \end{bmatrix} \succeq 0$ i.e. $M$ is positive semi-definite (PSD) given that $A$ is PSD and for some $\Lambda = \text{diag}(\lambda_1, ...
0
votes
1answer
31 views

How do I solve for A in the matrix equation $A - B(A./C) = D$?

I've got $A - B(A./C) = D$, and I want to solve for $A$.* $A$ is an unknown 2x1 vector, $B$ is a known 2x2 matrix, $C$ is a known 2x1 vector, and $D$ is a known 2x1 vector. *The notation $A./C$ ...
0
votes
0answers
20 views

Best Approximation In Inner Product Space (Polynomaly)

Question: Let $q_1=1, q_2 =t , q_3 = t^2-2$ {$q_1$,$q_2$,$q_3$} orthogonal basis of $P_2$. Find the BEST APPROXIMATION to the $p(t)=5- 1/2 t^4$ by the polynomial in $P_2$. Answers: ...
1
vote
3answers
53 views

Transposing matrix when differentiating it

Hi so I am trying to understand the solution of linear regression with matrices (found at the following link) and an confused about how on page 10 he says the derivative of $2Y'XB$ with respect to $B$ ...
3
votes
1answer
59 views

(Nonunique) Solvability of Sylvester Equation

I am interested in stating existence of solution of a Sylvester equation $$ AX - XB = C, $$ where $A$, $B$, $C$, and $X$ are $(n,n)$ matrices. Existence of a unique solution $X$ is given, if $A$ ...
8
votes
4answers
1k views

Matrix inverses - Why are they derived the way they are?

Note that this is not a question of how, but why. I know the mechanics of it, but this is the first thing i've come across that truly seems like magic, rather than a rigorous mathematical process. ...
0
votes
0answers
20 views

Eigenvalues of normalized adjacency matrix

Can anyone introduce some references on the eigenvalue estimation of normalized adjacency matrix, i.e., $W=D^{-1}A$ ($D$ is the degree matrix and $A$ is the adjacency matrix of the corresponding ...
0
votes
1answer
22 views

Properties or solution of C=I+wCw^T matrix equation?

In a project, I came to the following matrix problem: $$C_1=wC$$ $$C=I+wCw^\dagger$$ Where the unknown matrices are $C$, which is hermitian positive definite, and $w$, general not hermitian, no ...
-1
votes
0answers
11 views

Element-wise increasing series of matrices

Assume $X_{n+1} = aI(B - X_n C)^{-1}$, where $a$ is a positive scalar and $I$ is the identity matrix, $B$ is an M-matrix, and $X_n$ and $C$ are all-positive matrices. I am wondering if I can come up ...
0
votes
0answers
28 views

Equations system with unknown A matrix

It's been a while since I studied Linear Algebra, so forgive me if I'm inaccurate with terms. I have the usual system, which I can express in matricial form like $\mathbf{b} = A \mathbf{x}$. But A ...
2
votes
2answers
44 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
0
votes
0answers
20 views

Compact representation of the following matrix

I have a matrix that has the following structure \begin{bmatrix} a_1(1) & 0 & a_2(1) & 0 \\ 0 & a_2(1) & 0 & a_1(1) \\ a_1(2) & 0 & a_2(2) & 0 \\ 0 & a_2(2) ...
0
votes
2answers
26 views

Probability of two strings being equal

Given a matrix $A\in F_2^{n\times m}$, (let $m< n$ and $A$ has full column rank) what is the probability under the distribution ( $y,y'$ uniformly random in $\{0,1\}^m$), such that $Ay=y'$? I am ...
0
votes
1answer
31 views

How do I solve for vector $P$ in the matrix equation $s=A'B^{-1}A$?

I would like to rearrange the matrix equation $s=A'B^{-1}A$ into the form $A=f(s,B)$ (i.e., some function of $s$ and $B$), where s is scalar, $A$ is $n\times 1$, $A'$ is the transpose of $A$, and $B$ ...
1
vote
0answers
27 views

Differentiating quadratic form containing vector raised to powers elementwise, can I avoid Hadamard notation?

Say $\mathbf{M}$ is a symmetric, p.d. 2x2 matrix, and $\mathbf{x}$ is a 2x1 vector. The familiar quadratic form is of course given by: $A=\mathbf{x'}\mathbf{M}\mathbf{x}$ (where $A$ is a scalar), and ...
0
votes
1answer
30 views

existance of a solution to quadratic form equation

Let $\lambda$ is an unknown scalar and; $Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices, $B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors, $m=m_1 - ...
0
votes
0answers
26 views

Linear Operator on real (positive definite) symmetric matrix; Generalization of Lyapunov theorem

I am wondering if there is any results on a somewhat "generalization of Lyapunov theorem". By which I mean, as we know from Lyapunov theorem, for a Lyapunov operator on real symmetric matrix, $L_A: ...
3
votes
1answer
99 views

Describe the space of solutions to a simple matrix equation

I would like to describe the space of solutions to the following matrix equation. Here $U$ and $V$ are two unknown real matrices with $n$ lines and $p$ columns (i.e. they belong to ...
1
vote
1answer
40 views

Optimizing a function of a matrix

Let \begin{equation} \begin{aligned} W= & \underset{X}{\mathrm{maximize}} & & \log \left|X + K_1\right|- \alpha \log \left|X + K_2\right|\\ & \mathrm{subject \; to} & & 0 ...
2
votes
2answers
50 views

How to determine a function of a matrix is increasing or decreasing

We know that the derivative of a function can be used to determine whether the function is increasing or decreasing on any intervals in its domain. If $f'(x) > 0 $ at certain interval I, then the ...
1
vote
0answers
47 views

Solving a system of equations $A\vec{x}=\vec{b}$

I have the following matrix: $$A=\begin{pmatrix}1&2&1&6\\-2&3&5&6\\ 3&6&3&6\\-4&-4&0&23\end{pmatrix}$$ The question on my problem sheet asks me to ...
2
votes
4answers
66 views

Finding the rank of matrix $A^2$

supose $A$ is a $4\times4$ matrix such that $\operatorname{rank}(A)=4$. Find the rank of the matrix $A^2$. if there is a major rule for the power $k$ and not specially the power $2$.
1
vote
0answers
103 views

If $A$ is a $3 \times 3$ matrix. and $B=A'A$, then what can be said about the eigenvalues of $AB$?

If $A$ is a $3 \times 3$ matrix and $B=A'A$, then what can be said about the eigenvalues of $AB$? No form of $A$ is given; then how to proceed ? Can this problem be at all solved? If anyone can ...
0
votes
1answer
32 views

constructing a matrix such its square is not '0' but its cube is.

i have been asked to construct a matrix A such that $A^2$ is not equal to '0' but, $A^3=0$. how should i proceed. i can only understand that all the eigenvalues for A , $A^2$ and $A^3$ will be ...
0
votes
3answers
32 views

What is $[M_1,M_2]$ equal to? ($M_1$ and $M_2$ are matrices)

This is an old exercise that I had a year ago: $$M_1 = \dfrac{1}{\sqrt{2}} \begin{bmatrix}0 & 1 &0\\1 & 0 & 1\\0 & 1 & 0\end{bmatrix}$$ $$M_2 = \dfrac{1}{\sqrt{2}} ...
0
votes
1answer
23 views

How to solve for matrix $X$ in $Y=X(X^TDX)^{-1/2}$

Let $Y \in \mathbb{R}^{n \times n}$ be any matrix such that $Y^T D Y = I$ for some positive diagonal matrix $D$ and $I$ the identity matrix. Further it is known that $Y=X(X^TDX)^{-1/2}$ for some ...