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

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1answer
14 views

optimize matrix equation (low rank)

I am trying to solve the following optimization problem $min_{H,V}$ $||A-HV||_F^2$ s.t $V\geq 0$ (i.e all entries are non-negative) and H is low rank Is there a way to tackle the is problem Can ...
0
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0answers
13 views

optimization query

i am trying to optimize the following matrix function $\min_{H,V}$ $\|A-HV\|_F^2$ under the following constraints (1) $H$ is low rank (2) $V$ is positive (all entries of $V$ are non-negative) Any ...
0
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0answers
8 views

Expansion of terms in matrix equation

If $R,G$ are $n \times 1$ vectors, and $S$ is an $n\times n$ matrix. How can I expand: $$ (R-G)^TS(R-G) $$ where $T$ denotes the transpose. Can I write it out as: $$ =R^T S R - R^T S G - G^T S R + ...
0
votes
1answer
29 views

Matrix polynomials/eigenvalues

$\begin{pmatrix} 7 & -2\\2 & 2 \end{pmatrix}$ The eigenvalues for this matrix are $\lambda=6$ and $\lambda=3$ It also happens that $(A-6I)(A-3I)=0$ I've checked for various $2$ x $2$ ...
0
votes
2answers
29 views

Approximate solution to a matrix equation

Let $A$ and $B$ be $n \times m$ matrices. I am looking for a $m \times m$ matrix $X$ which would be an approximate solution to the equation $AX = B$ (an exact solution is very unlikely to exist). More ...
0
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2answers
23 views

Finding eigenvalues from characteristic polynomial

I am finding it extremely hard to find the eigenvalues after finding the characteristic polynomial. For example (instead of $\lambda$ I will use $x$) I have: $-x^3+x^2+16x+20=0$, how do i find the ...
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2answers
26 views

questions about systems of equations using matrices and row echleon

I have the following matrix: $$ \left[ \begin{array}{cc|c} -1&-2&{\sqrt 2}\\ -8&2&{\sqrt 3} \end{array} \right] $$ So the first thing I do is multiply R1 by - 1 to ...
0
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0answers
6 views

How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$

How to find $B$ by solving the following linear system: $s_k$ $B$ ${s_k}^T$ $=1,$ $\qquad$ for $k=1 ... ,p$. Where $s_k$ is a $1\times3$ row_vector from the matrix $S= [s_1 ... ...
0
votes
0answers
25 views

Find a relation between a,b and c

$ a,b,c\in \Bbb R$ $2x_1+2x_2+3x_3=a$ $3x_1-x_2+5x_3=b$ $x_1-3x_2+2x_3=c$ if a,b and c is a solution of this linear equation system find the relation between a,b and c I dont understand the ...
1
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1answer
8 views

Change of Basis in Canonical Correlation Analysis

I am studying canonical correlation analysis. And I'm completely stumped for the last few days at the following manipulation. How does the following change of basis works? The equation doesn't even ...
1
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2answers
40 views

How can you expand the adjoint of a matrix into a polynomial with matrix coefficients?

This book contains an algorithm which claims that a matrix $sI - A$, where $A$ is some $n \times n $ square matrix and $s$ a variable can be expanded into $$adj(sI - A) = K_0 s^{n-1} + K_1 s^{n-2} ...
2
votes
2answers
76 views

How to derive this matrix equation

$$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r} \right) \right\| }_F^2 $$ I think $ ...
3
votes
2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
1
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2answers
50 views

What are the Eigenvectors in the following matrix?

I have the matrix A: \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4\\ \end{bmatrix} I found $\lambda I_n - A$ to be: \begin{bmatrix} (\lambda -4) & -2 & -2\\ -2 ...
3
votes
2answers
69 views

Proof of multivariate regression plane maximizes correlation in normals

I am doing a homework sheet as practice for an upcoming course in multivariate statistics and been stuck on the following problem: Let ...
1
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1answer
19 views

Matrix equation problem

Is my solution of this matrix equation correct ? $(AX^{-1}+B)^{-1}=\frac13X$, I've started off by raising the equation to $-1$ and in the end I get : $((\frac13I-A)^{-1}B)^{-1}$
0
votes
1answer
34 views

Help Finding Elementary Matrix.

I apologise in advance if this is simple, but I'm losing my brain over this question. I'm unsure how to make the matrix format work either. I'm trying to find elementary matrices so that ...
0
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0answers
18 views

Help for solving this optimization problem

Are given $2$ square matrices $M_1$ and $M_2$ of dimension $d \times d$ and two points in a $d$-dimensional space $p_1$ and $p_2$ ($d \times 1$). Now I need to find two other square matrices $X$ and ...
-2
votes
1answer
22 views

How many solutions depending on the parameter (augmented matrix?)

I have to find how many solutions have got the following equations, depending on p parameter? $ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ ...
0
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2answers
34 views

Is there a linear decomposition of the Hadamard inverse of the sum of two matrices?

Let the matrix $$\Gamma = \alpha A + (1-\alpha)B$$ where $B$ is a square symmetric matrix, $A = c\ ee'$, where $e$ is a vector of ones, and $c$ a positive constant and $0 < \alpha < 1$. The ...
1
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1answer
29 views

Can a matrix M be written as A*B*C where A and C are symmetric positive definite?

Given a matrix $M$ and $B$ (not necessarily square), is there a way to determine whether symmetric positive definite matrices $A$ and $C$ exist such that $M=A B C$ ?
0
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1answer
18 views

Spectral decomposition of idempotent matrix

Let $C$ be an idempotent and symmetric matrix and its spectral composition is given by $$C = ADA^T$$ I cannot see how you can rewrite this to be $$D = A^TCA$$ I've found this reformulation in a book ...
6
votes
3answers
130 views

Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$

I have the following question : Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$ I managed to proof that $I+BA$ invertible My proof : We know that $AB$ and $BA$ ...
0
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0answers
22 views

How to derive one matrix algebra equation from another

I have one matrix equation: s = A' X^-1 A (where s is scalar, A is a vector, A' is its transpose, and X^-1 is the inverse of a square, symmetric matrix) Which can be transformed into another ...
0
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0answers
28 views

trivial and non-trivial solution of homogeneous equation

Suppose I have system of 3 equations $$a_1x+b_1y+c_1z=0$$ $$a_2x+b_2y+c_2z=0$$ $$a_3x+b_3y+c_3z=0$$ and cofficient matrix $A=\begin{equation} \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 ...
2
votes
2answers
28 views

Are these statements equivalent?(invertible matrix theorem)

Suddenly I got a question about the invertible matrix theorem. Among lots of equivalent statements suggested in my lin-alg text, I'm confused whether the statement that 'The equation Ax=b has "at ...
0
votes
2answers
26 views

Check if solution to a system of linear equations is positive

If I have a system of linear equations, $$\mathbf{A}\mathbf{x}=\mathbf{b},$$ where $\mathbf{A}$ is a $n \times n$ matrix with real coefficients and $\mathbf{b}$ a vector of size $n$ with real ...
0
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1answer
38 views

gauss jordan matrix involving parameter $k$

Could anyone help me in solving this matrix? $$\left[\begin{array}{ccc|c} k+2& k-1& k& 2\\ 0& k+2& 2& 0\\ 0& 0& k^2+k-2& k+2 ...
2
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1answer
25 views

Solving inhomogenous continuous-time system with non-diagonalisable system matrix

I have an exercise where i have to find the general solution to this problem: $$ X'=\left( \begin{matrix} 2&-1\\ 4&-2 \end{matrix} \right)X + \begin{pmatrix} 2\\1 \end {pmatrix}. $$ ...
0
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1answer
24 views

how to insure integral coefficient linear equations has a unique integral solution

I have a problem. I need to solve an integral coefficients linear equations with $m$ number of equations and $n$ number of variables, and $m<n$. The coefficients can be defined by myself, so 1.I ...
1
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1answer
45 views

Prove $\begin{pmatrix} a&b\\ 2a&2b\\ \end{pmatrix} \begin{pmatrix} x\\y\\ \end{pmatrix}=\begin{pmatrix} c\\2c \\ \end{pmatrix}$

Prove that $$ \begin{pmatrix} a & b \\ 2a & 2b \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} c \\ 2c \\ \end{pmatrix} $$ has solution $$ \begin{pmatrix} x \\ y \\ ...
-1
votes
3answers
38 views

Matrix Equation? Maybe showing $AB=I_n$ works. [closed]

Let $A, B$ be two $n*n$ matrices with the property that $ABX = X$, $\forall X$ a matrix $n*1$. Prove that $A$ and $B$ are invertible matrices.
0
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1answer
28 views

How to solve this matrix equation? Does the solution have a closed-form?

I am trying to find solution $X \in SO(3)$ for this matrix equation. $$PX - XQ + YXZ = K$$ where matrices $P, Q, Y, Z, K \in\mathbb{R}^{3\times3}$ are known. At the first glance, it does not seem to ...
0
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0answers
69 views

How to solve series of 8 equations with 8 unknowns?

In this article http://www.fmwconcepts.com/imagemagick/bilinearwarp/FourCornerImageWarp2.pdf they speak of solving for a0,a1,a2,a3,b0,b1,b2,b3 but I want to know ...
1
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1answer
51 views

Conditions to preserve Laplacian matrix

Let $L$ be a Laplacian matrix $L=D-A$ where $D$ and $A$ are the degree and the adjacency matrices. It is known that $L$ has (among others) the properties: $L=L^T$, $L\geq 0$ and $L1_n=0$, where $1_n$ ...
0
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3answers
32 views

Question on matrices

Let A be a symmetric matrix of form 2.2 then what should be the elements in matrix B of2.2 such that AB not equals to BA
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0answers
51 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
0
votes
1answer
93 views

Calculating matrix derivatives with MATLAB or MATHEMATICA?

I'd like to calculate the following derivative, \begin{equation} \frac{d\|(f(C)\cdot f(C)^{+}-I)\cdot u\|^2)}{dC} \end{equation} Where $C$ is a matrix of dimension $n\times k$ (s.t $k < n$). And ...
0
votes
1answer
23 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
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1answer
41 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 ...
2
votes
1answer
47 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
168 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
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1answer
15 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 ...
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0answers
52 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
50 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
40 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
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1answer
28 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
25 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
38 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
44 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} = ...