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

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0answers
19 views

Using semidefinite programming to solve the following problem

I am struggling with the following problem, and wonder is SDP can help: $$\mathrm{maximize\ } \alpha_{10}+\alpha_5+(\alpha_2+\alpha_8)/2 \mathrm{\ subjected\ to\ } \mathrm{T_1}\succeq0, ...
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1answer
30 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
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1answer
23 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
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1answer
27 views

How can I translate this problem into Matrix/Linear Algebra notation?

I have a matrix H of size s×d with holdings of s stocks across d days. H shows how many shares of each stock is in my portfolio on each day. The number of shares can change from day to day due to ...
1
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2answers
39 views

Finding a matrix from equation

we've got the following 4x4 Matrix $$\begin{pmatrix} 4 & -2 & 3 & 2\\ 3 & 5 & 1 & -4\\ -1 & 6 & -4 & -7\\ -2 & 0 & -2 & 4 \end{pmatrix}$$ and I need ...
1
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1answer
23 views

Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[m,n]$ and $b$ is $[1,n]$ matrix. These all ...
0
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1answer
80 views

Find the values of a and b such that the sytem has a unique solution and a two-parameter solution?

\begin{bmatrix} a & 0 & b & 2 \\ a & a & 4 & 4 \\ 0 & a & 2 & b \\ \end{bmatrix} Find the values of a and b such that the system ...
1
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1answer
17 views

Degree $2$ nilpotent matrices with non-zero product and subset products not depending on the order

Follow up to this answered question. Let $n$ be sufficiently large positive integer. Let $S=\{M_i\}$ be a set of $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/m \mathbb{Z}$. ...
3
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2answers
45 views

Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.

I have some questions regarding the following problem Let $ A + iB $ - hermitian and positive definite, where $A, B \in \mathbb R^{n\ \times\ n} $ show that the real matrix $$C =\begin{pmatrix} A ...
1
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1answer
35 views

Relation between Hermitian Matrices

For a given Hermitian matrix $A$, what is the semidefinite positive matrix $B$ such that $$\mathbf{y}^{H}B\mathbf{y} = \left | \mathbf{y}^{H}A\mathbf{y} \right |, $$ for all $\mathbf{y} \in ...
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1answer
42 views
+50

A problem with unitary matrices

Let $U$ be an $n \times n$ unitary matrix. And let $|\cdot |^2: \mathbb C^n \rightarrow \mathbb R_+^n$ be the function such that $|\mathbb w|^2$ is the vector which has its $i$-th entry equal to ...
2
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3answers
60 views

Find the inverse of a submatrix of a given matrix

I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ ...
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2answers
25 views

Finding reverse matrix from equation

we have the following matrices: ...
0
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1answer
18 views

Solving Square matrix equations

A is a Square matrix, and we have the following function t(A)=Σ(i=1 to n)a(ii) it equals to the sum of the diagonal elements of the matrix. Prove that for every ...
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0answers
9 views

Solutions to a laplacian preservation

I'm trying to write an impementation to that paper over here http://www.cs.jhu.edu/~misha/Fall07/Papers/Sorkine04.pdf The main idea is that i have a series of points, and i displace some of them. ...
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1answer
12 views

proving equation of an invertible matrix

as far as i can tell the following sentence is true but what are the steps to actually prove it? ...
0
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1answer
16 views

Invertible matrix equation

I am trying to prove OR to rule out the following sentence and i'm kind of stuck. if A,B are Invertible matrix, then A+B is also an Invertible matrix? what are the steps to prove OR to rule it ...
3
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0answers
51 views

Number of solutions of an equation

Fix a vector $x\in\{0,1\}^n$, and let $a$ be a random vector in $\mathbb{Z}^n_q$ for some prime $q$. Consider $y=ax$, and $S=\{x'\mid ax'=y\}$. I want to compute the probability that $\lvert S ...
0
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2answers
47 views

Differentiation of a Vector with respect to a vector

I am studying a paper and I am going crazy about one differentiation which it is written on it but not explained. I think I am missing something and probably something easy. I would love if someone ...
1
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0answers
42 views

When is the product of two arbitray matrices symmetric?

Let $\mathbf{A}$ be a real $n \times m$ matrix. Let $\mathbf{B}$ be a real $m \times n$ matrix. How to solve the following matrix equation? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$ ...
2
votes
3answers
60 views

Clockwise rotation of $3\times3$ matrix?

I've recently been studying matrices and have encountered a rather intriguing question which has quite frankly stumped me. Find the $3\times3$ matrix which represents a rotation clockwise through ...
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2answers
38 views

Calculate matrix $X$ in expression $X + B = (A-B)X$

I have to calculate matrix $X$ in expression $X + B = (A-B)X$. $$ A=\left[ \begin{array} k1 & -2 & 3\\ 2 & 4 &0\\ -1 & 2 & 1\\ \end{array} ...
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0answers
18 views

Transformations and matrices

Anyone able to solve this? If λ=37.115 then If λ=10.885 then Then we need to diagonalize matrix A by expressing it in the form A=PDP-1 Choosing -13 13 so D= 10.885 0 ...
1
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1answer
29 views

Can independence of a system and a vector be establish if there is only cross-indepedence?

Say that I have the following linear system: $$[A a'] \begin{bmatrix} x \\ x' \\ \end{bmatrix} =Ax + a'x' $$ I want to know when this system is zero if and only if ...
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3answers
49 views

Determine 9 variables by 3 equations with approximation

I have an equation in the form of Q*d=z, where Q is 3by3 matrix of variables, and d and z are vectors of 3 known numbers. What would be the best way to compute all 9 elements of matrix Q, provided ...
2
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0answers
39 views

Let A be an m × n matrix, and b an m × 1 vector, both with integer entries.

Let $A$ be an $m \times n$ matrix, and $b$ an $m \times 1$ vector, both with integer entries. If $Ax = b$ has a solution over $ \mathbb Z/p \mathbb Z $ for every prime $p,$ is a real solution ...
0
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0answers
11 views

there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range?

Let $X,Y \in {M_{n \times m}}$ have orthonormal column. Also there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range(column space) ?
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1answer
13 views

$X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Let $X,Y \in {M_{n*m}}$ have orthonormal column. Also $X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?
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0answers
9 views

Can this equation have an explicit solution?

Given $n > 0$, $0 \leq i \leq n$ is an integer, $D = diag(d_1, \dots, d_n)$ is positive definite, $e_i$ is the $i$th column of a $n \times n$ identity matrix, $u \in R^n$ such that $B = D + u * ...
0
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2answers
46 views

Inverting matrix multiplication “and” representing with a smaller sized matrix

Consider I have a vector $A=[a_0 \ \ a_1]$ and a random binary matrix $B$ which is $2\times 2$. I compute $C=A\cdot B$. My question is: " Can one compute $B$ Given $C$ and $A$? " Note: By binary ...
2
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3answers
33 views

$n \times m$ matrix conversion?

Is it possible to convert an $n\times m$ matrix $A$ such that $$ A=CB $$ where $B$ is a $1\times m$ matrix which contains all elements of $A$, and $C$ is a $n\times 1$ matrix. I'm assuming no since ...
2
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1answer
48 views

The number of linearly independent solution of the homogeneous system of linear equations $AX=0$

I came across the following multiple choice question: The number of linearly independent solution of the homogeneous system of linear equations $AX=0$, where $X$ consists of $n$ unknowns and $A$ ...
0
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0answers
19 views

Generate duplicate element from a matrix by formula $b(i:j)=A(i:j,:) \times A^{-1} \times b$

I have an interesting question about generate duplicate elements from matrix. I assume that I have a matrix A (such as the bellow example $5 \times 5$) and vector $b$ is $5 \times 1$. My goal is make ...
0
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0answers
35 views

Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S How would one go about working S^-1/2 (inverse of the principle square)? Bearing in mind, I'm trying to understand a paper which states: I've ...
1
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2answers
18 views

Scalar versus linear equation of a plane

What is the difference between a scalar and a linear equation of a plane? In my textbook it says that a scalar equation is $a(x-x_1)+b(y-y_1)+c(z-z_1)=0$ and a linear equation is $ax+by+cz=d$ How do ...
0
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1answer
48 views

Matrix inverse and Hermitian transpose

Could anyone help me to prove the following the equation? $\large ( G_2^H G_2 + K_w^{-1} )^{-1} = Q$ which leads to $\large K_w = Q - Q G_2^H ( G_2 Q G_2^H - I )^{-1} G_2 Q$ Here ...
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1answer
52 views

How to figure out how many entries are in a relation

I have the domain $A = \{1, 2, \ldots , 1000\}$. I need to figure out how many non zero entries are in each relation: a. $R_1 = \{\;(a, b) \;|\; a \le b\;\}$ b. $R_2 = \{\;(a, b) \;|\; a + b = ...
0
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0answers
28 views

Fast algorithm to invert a large sparse matrix

I am interesting in sparse matrix that defined at here. I am looking for a fast algorithm to invert the matrix (better than Gaussian Elimimation). Could you suggest to me some methods that reduce ...
1
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1answer
34 views

Convert a general equation system to matrix form

I have an equation, where $x$ is a vector of variables and the rest (vector $a$, vector $b$, matrix $M$ and $c_1, c_2, c_3$) are parameters: $a_i - x_i + c_1\sum_{j=1}^{n} m_{ij} x_j - c_2 ...
0
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0answers
9 views

Dimensions of matrices around a diagonal matrix?

The matrices $\mathbf{L}$, $\beta$ and $\mathbf{c}$ are ($j \times b$), ($b \times 1$) and ($j \times 1$) dimensional, respectively, with $j \le b$. The matrix $\mathbf{X}' \mathbf{X}$ is a diagonal ...
1
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1answer
22 views

Finding a solution to matrix equation occurring inside an optimization problem

As a part of an optimization problem (while equating the derivative of the cost function to 0), I'm getting the following expression. $$-2XX^TC + 2XX^TACC^T + \gamma GA = 0,$$ where, $X, C, G$ are ...
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0answers
47 views

a special matrix inverse

Let $A=\left( \begin{matrix} {{A}_{11}} & \ldots & {{A}_{1n}} \\ \vdots & \ddots & \vdots \\ {{A}_{n1}} & \cdots & {{A}_{nn}} \\ \end{matrix} \right)$ be an ...
0
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1answer
15 views

Computing S*S' and comparing to eye(5)

In matlab, we asked to set A=rand(5,2)*rand(2,5) then to set Q=orth(A), W=null(A'), ...
1
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0answers
33 views

Using Cholesky factorization to solve the system AXA=B

I have been given a problem of solving X, which is an unblurred image, in the system: $$B = A X A \iff X = A^{-1} B A^{-1}$$ Where the matrix A describes the blurring of an image and the matrix B is ...
0
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2answers
45 views

matrix product equals 0

Be $A,X$ two matrices of the same order $n$ . Find the necessary and sufficient condition for $A$ so that there exists $X$ with the property $AX=XA=0_n$. I believe that rank A =1 is the condition ...
0
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0answers
14 views

Matrix algebra with Cadabra

I am novice in mathematics. I need to do a lot of symbolic matrix algebra. I found that Cadabra have a lot of facilities to do tensor calculations. But I need only matrix algebra. Is there a better ...
0
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1answer
32 views

Suppose A is an mxn matrix, if all b is in R m (i.e. A spans R m), is A necessarily independent?

I think the answer is No. Am I missing something here?: Note: What I'm saying is that if A is an mxn matrix that is Linearly Independent, A MUST span R m. However, if A spans R m, A isn't necessarily ...
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2answers
49 views

How to represent matrix as the sum of rank-one matrices

If we're given $B$ to be a $4 \times 7$ matrix: $$\begin{bmatrix}1 & 2 & -3 & 7 & 0 & -2 & 5\\1 & 2 & -3 & 7 & 1 & 3 & -2\\0 & 0 & 0 & 0 ...
0
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1answer
24 views

How to set up matrix to compute best coefficients

Suppose we're given a non-linear spring with the following relationship between the applied weight ($x$) and displacement ($y$): $y = ax + bx^3$. I've done a sequence of $m$ tests measuring the ...
1
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2answers
33 views

If matrix $P$ is positive semidefinite, is $C^TPC$?

Or if it's a conditionally true statement, what must be true about the matrix $C$ to make $C^TPC$ PSD? edit - all matrices are made of real numbers. edit - here is the issue I'm running into that is ...