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

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-1
votes
0answers
10 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
23 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
42 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
18 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) ...
-4
votes
0answers
40 views

Proving linear algebra equation [on hold]

I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent. (This problem comes from Bayesian estimation, ...
0
votes
2answers
23 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
29 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
22 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
25 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
19 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
96 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
39 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
44 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 ...
0
votes
0answers
17 views

Calculate left and right null vectors of a 3x3 matrix using SVD

I have a $3\times3$ Matrix $T$ and need to calculate the left and right null-vectors $u_i$ and $v_i$, such that $$u_i^TT = 0,\qquad Tv_i = 0.$$ When I calculate an singular value decomposition of ...
2
votes
4answers
64 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
101 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 ...
-1
votes
0answers
12 views

Codechef-June15 Problem Based on Binary Matrix: Study on System of equations: A → B = ¬ O and B → A = O

Question link: http://www.codechef.com/JUNE15/problems/SEADIVM Binary Matrix: Study on System of equations: $A \to B = \lnot O$ and $B \to A = O$,i.e. logical operations on boolean matrices $A\to B$ ...
0
votes
1answer
27 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 ...
-1
votes
0answers
38 views

Finding a basis for $R^2$ with some constraints

Find a Basis $B$ of $R^2$ s.t. (1) $\left(\begin{array}{c}1 \\2\end{array}\right)_B = \left(\begin{array}{c}3 \\5\end{array}\right)$, and (2) $\left(\begin{array}{c}3 \\4\end{array}\right)_B = ...
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 ...
9
votes
1answer
80 views

Find the matrices $A\in M_{n}(\textbf{Z}/n\textbf{Z})$ such that $A^3=I_{n}$

If $n\in\textbf{N}$, what are the matrices $A\in M_{n}(\textbf{Z}/n\textbf{Z})$ such that $A^3=I_{n}$ ? An exercise I found and have no clue how to do. Even if I assume $n$ is prime and we are ...
1
vote
1answer
67 views

Does the matrix equation $\text{A}^k=\epsilon \text{A}$ have any $\text{A}\neq I_n$ solutions?

The question was recently posed to myself and some peers of whether or not the equation $$\text{A}^k=\epsilon\text{A}\ \Big|\ k, \epsilon \in \mathbb{R}, $$ where A is $n\times n\ \big|\ n>1,$ has ...
0
votes
1answer
21 views

Gradient of a forth order scalar function with respect to a Matrix

I'm trying to take the gradient of the following function w.r.t A: $$ f(A) = ||AC_YA^T - C_R||_F^2 $$ I tried the following: $$ f(A) = trace((AC_YA^T - C_R)^T(AC_YA^T - C_R)) = ...
1
vote
0answers
29 views

Technique to solve 2 x 2 block Toeplitz system

I want to know how to solve this system of equations: $$ \begin{bmatrix} R_{N} &-Q_{NM} \\ -Q_{NM}^T & P_M \end{bmatrix} \begin{bmatrix} a_N \\ b_M \end{bmatrix} = \begin{bmatrix} ...
1
vote
1answer
90 views

Matrix equation solving

I have these matrices given: $$A= \begin{bmatrix} 3&1&1&1\\1&2&0&{-1}\\0&1&2&{-1}\\0&0&1&1\end{bmatrix}$$ $$T= \begin{bmatrix} ...
0
votes
3answers
41 views

Find a base in linear algebra

So I have this problem: Let $V_1 = \{(x_1,x_2,x_3) \in \mathbb R^3 : x_1+x_2-x_3 = 0\} $ and $V_2 = \{(x_1,x_2,x_3) \in \mathbb R^3 : x_1-2x_3 = 0\} $. Find a base in $V_1∩V_2$ and show that $V_1+V_2 ...
2
votes
2answers
29 views

Matrix roots of the characteristic equation

Let A be a matrix of $n \times n$ dimensions and $p( \lambda)= \det (A- \lambda I)$. Then $p(A)=0$ by Caylee-Hamilton. Are there any other matrices that satisfy the characteristic equation of A?
0
votes
1answer
42 views

solution for Matrix equation

$$ (w*(R_1*P*R_1^{-1})^{-1}+w*(R_2*P*R_2^{-1})^{-1})^{-1}=P_{th} $$ $R_i$ is a rotation matrix 2*2: $$ R_i=\left[\begin{matrix} cos\theta_i & sin\theta_i \\ -sin\theta_i & ...
0
votes
0answers
16 views

Will these two multiplications lead to the same result?

Before you say that this is a statistics question, bear with me for a moment. This is more about transformation matrices and multiplication. So let's say that I have a matrix of N-dimensional ...
0
votes
0answers
13 views

how can matrix multiplication be described in traffic cordinations?

In one of our lectures, spacial Interaction was represented as matrices. but when It's come to multiplications how would you merge spacial interactions with the result of multiplication. To make ...
0
votes
3answers
32 views

Prove that $A$ is regular matrix and find $A^{-1}$ if $A,I\in M_{n\times n}(\mathbb{R})$ and $(A+I)^3=O$

Expansion of a binomial gives: $$A^3+3A^2+3A+I=0$$ We know that the matrix is regular if squared and has inverse ($detA\neq 0$). Is it possible to determine from above equation that $detA\neq 0$?
1
vote
2answers
43 views

Prove that $A^{-1}=A^n$ if $A^n+A^{n-1}+…+A+E=O,n\in \mathbb{N}$ and $A$ is regular matrix

I took the example of $2\times 2$ matrix $$ \begin{bmatrix} x & y \\ u & v \\ \end{bmatrix}$$ which gives matrix equation: $$A^2+A+E=O$$ After addition, I get system ...
0
votes
5answers
142 views

$AB-BA=A$, Then A is singular? [duplicate]

Title is the question, I tried taking trace both side and got trace of $A$ is zero, now to conclude $A$ is singular, suppose $A$ is non singular, then multiplying both side by Inverse of $A$ we get ...
1
vote
3answers
44 views

Book on linear algebra containing interesting problems

Could anyone suggest me a problem book on linear algebra that contains interesting problems on rank, nullity, nullspace, linear transformations, eigenvalues, eigenvectors and characteristic ...
0
votes
0answers
26 views

interpreting negative square root of a matrix

I am working on stability analysis of systems with impacts and in my algorithm, I have reached a state where I have $xPx = 1$, where $P$ is a positive semi definite matrix. now with my code, at a ...
2
votes
1answer
79 views

Is there a way to directly solve this matrix equation: $XAX^{T} = B$

$X^{T}$ is the transpose of $X$. $A$ is a $n$ x $n$ matrix and $B$ is a $m$ x $m$ matrix, $m$ > $n$, both of them are known, $A$ is positive definitive and $B$ is symmetric. I would like to find $X$. ...
1
vote
1answer
42 views

Linear combination to recover particular data entry from denoised data?

Let $\mathbf{x} = [x_1, x_2, x_3]^t$ the 'data' where $x_1$ is considered to be 'noise', $M$ a $3\times 3$-matrix with full rank, and $\mathbf{y} = M\mathbf{x}$ the obserced mixture. Let $m^-_i$ ...
0
votes
0answers
16 views

Help finding the critical values of α where the qualitative nature of the phase portrait for the system changes?

I was asked to solved for the eigenvalues in terms of α for 2X2 matrix and so i did and my answer was marked as correct. Then I was asked to solve for this: The roots are complex when? There is a ...
5
votes
1answer
34 views

Linear Transformation Matrix with polynomials

A linear transformation $T : P_2 \to P_2$ has matrix with respect to $S$ given by: $$[T]\,( S) = \begin{bmatrix} 1/2&-3&1/2\\ -1&4&-1\\ 1/2&2&1/2\\ \end{bmatrix} $$ How do ...
5
votes
1answer
49 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
1
vote
1answer
34 views

What are the facts used in each step of this proof?

What are the facts used in each step of this proof ? Suppose that $A\in F^{nm}$ and $B\in F^{ml}$ $$\begin{align}rank A + rank B &= rank\begin{bmatrix}0 & A\\B & 0\\ \end{bmatrix}\\ ...
0
votes
0answers
6 views

Determining pitch and roll angles from the coordinates of a vector

I want to know, given the measurement of an accelerometer at rest (so not really an acceleration but a force per unit of mass) the inclination of this accelerometer, along the X and Y axis. So, In ...
0
votes
1answer
49 views

Proof: $ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$

I am looking for a proof of the following:$$ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$$ For positive hermitian matrices A and B, and a unitarily invariant norm $ |||\cdot|||$.
1
vote
1answer
37 views

Let A and B be n*n matrices such that trace(A)<0<trace(B).

Let A and B $n\times n$ such that trace(A)$\lt0\lt$trace(B). Then, $f(t)=1-det(e^{tA+(1-t)B})$ has 1) infinitely many zeros in $0\lt t\lt1$ 2) at least one zero in $\Bbb R$ 3) no zeros 4) either ...
2
votes
2answers
77 views

Solve matrix equation $e^A=e^B$ for nilpotent $A, B$.

I need to solve equaton $e^A=e^B$ for nilpotent matrices A and B over field $\mathbb C$, where $B$ is fixed. I solved equation $e^X=E$ for all matrices. The solution is any semisimple (in case ...
0
votes
0answers
32 views

How to make use of symmetric of sparse matrix to solve this kind of problem?

I have the following matrix to be solved: $$\left\{ \matrix{ {a_{11}}{x_1} + {a_{12}}{x_2} + \cdots + {a_{1n}}{x_n} = {y_1} \hfill \cr {a_{21}}{x_1} + {a_{22}}{x_2} + \cdots + {a_{2n}}{x_n} = ...
0
votes
1answer
40 views

Express a second-order cone (SOC) inequality as a linear matrix inequality (LMI)

For $y \in \mathbf{R}^n$ and $t \in \mathbf{R}$, show that: $$||y||_2 \leq t ~~\iff~~ F(y) \succeq 0$$ Where $\text{I}$ is the $n \times n$ identity matrix, and $$F(y) = \begin{pmatrix} t ...
2
votes
1answer
80 views

Proof: $|||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| $?

I am looking for a proof of the following: \begin{equation*} |||AB|||^ \frac{1}{2} \leq |||A ^ \frac{1}{2}B ^ \frac{1}{2}||| \end{equation*} Where A, B are positive, hermitian matrices, and $|||⋅|||$ ...