Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrix-equations]

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

0
votes
0answers
33 views

Solving a second order matrix differential equation [duplicate]

How to solve $$x''+2Ax'+A^2x=0$$ where $A$ is a constant matrix and $x$ is a vector ? I assumed the form was $x=c\cdot \exp(At)$ similar to how it does done for normal second order equations. I got ...
6
votes
0answers
102 views

If $AB=A$, does B have to be the identity matrix?

Suppose $A$ and $B$ are square matrices and that $AB=A$ with $B \neq I$. What does this say about the invertibility of $A$? This question showed up on an exam I took this past spring. I got stuck on ...
2
votes
0answers
52 views

How do I solve $\mathbf{x}^\top e^{t\mathbf{A}}\mathbf{y}=c$ for $t$?

I have the following matrix equation: $\mathbf{x}^\top e^{t\mathbf{A}}\mathbf{y}=c$ $\mathbf{x}$ and $\mathbf{y}$ are vectors of length $k$, $\mathbf{A}$ is a $k\times k$ matrix, and $c$ and $t$ ...
0
votes
1answer
30 views

Can i multiply an [6x1] matrix with an [6x6] in linear algebra?

I wish to multiply the first matrix float X[6] = { x, //Position y, //Position z, //Position _x, //Velocity _y, //Velocity _z //Velocity }; with ...
0
votes
0answers
19 views

Specifying a base $B_R$ for the row space and a base $B_C$ for the column space.

The following matrix over $\mathbb{Z_5}$ is given: $$ \begin{bmatrix} 2 & 2& 2& 3\\ 1&3&1&3\\ 3&0&2&2\\ 4&1&0&4\\ 1&2&2&0 \end{bmatrix} $$...
3
votes
3answers
39 views

Find the values of a and b that make the equation system have infinite solutions

I'm given the following system of equations and I need to find the values for a and b so that the system has infinite solutions. \begin{cases} &3y &- &2z &= 5 \\ &-2x &+ &...
0
votes
1answer
29 views

Factoring out a vector from a matrix product

I have the following expression $$ x^\top + a x^\top B = 0$$ where $a$ and $x$ are column vectors and $B$ is a matrix. Dimensions are such that matrix products work everywhere. Now my question ...
0
votes
0answers
20 views

Matrix perturbation and eigenvector

$Ae=\lambda e$ $e^Te=1$ $A$ is a real matrix. $\lambda, e$ are real. $(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$ Neglecting small terms, $\Delta Ae +A \Delta e=\Delta \lambda ...
0
votes
1answer
47 views

States of the world/Game theory and Beliefs

This post consists on 3 parts: the question itself, hint and a table. The question will make sense to you only after you have read the tables and the hint attached. The problem is about beliefs of a ...
0
votes
2answers
22 views

What is meant by tridiagonal linear equation system?

I have to implement the SOR (Successive Over-Relaxation) method, using sparse matrices, to find the solution vector of these linear equations systems (for quite huge matrices): What does that tridiag(...
1
vote
1answer
13 views

Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors

While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals ...
0
votes
1answer
14 views

Proving a specific $min$ function is equivalent to solving $Ax-b$

The homework question asks to prove that $min_{x\in\mathbb{R}} {f(x) = 1/2<Ax,x>-<b,x>}$ is equivalent to solving a linear system $Ax-b$. The hint the professor gave is to recite the ...
0
votes
0answers
28 views

How to solve a matrix dominated by zeros?

I am trying to solve a matrix of this form: Is there a known algorithm or a method to solve this kind of matrices more efficiently than a normal Gauß elimination method? I input the diagonals as ...
-2
votes
1answer
54 views

An $n \times n$ matrix $A$ is called skew-symmetric if $A^T = −A$. What values of a, b, c, and d now make the following matrix skew-symmetric? [closed]

Let $$ A=\left( \begin{matrix} d & 8a-c & 8a+2b \\ a & 0 & 8-5d \\ a+5b & c & 0 \\ \end{matrix} \right) $$ Let $$ A^T=\left( \begin{matrix} d & ...
2
votes
2answers
27 views

Quadratic form vanishing at certain points

Let $A\in\mathbb{R}^{d\times d}$ be a symmetric matrix, and $X_1,\dots, X_n\in \mathbb{R}^d$ be vectors with $n>d$ (if more convenient, one can assume ${\rm span}(X_1,\dots,X_n)=\mathbb{R}^d$. ...
4
votes
4answers
95 views

Given $A^2 = A - I$ find $A^{15}$

We should find matrix A such that $A^2 = A - I$ find $A^{15}$. I solved this with observing the pattern with raising A to different powers up to 6 and I realized that $A^{15} = -I$. However I'm not ...
0
votes
0answers
47 views

Show that a specific matrix difference is positive semidefinite

I need to show that for two specific matrices $A$ and $B$ the difference $$A-(I-B)A(I-B)^\prime$$ is positive semidefinite. How do I start this?
3
votes
1answer
71 views

Necessary conditions for positive realness

Given a linear time-invariant (LTI) system with $$ \begin{align} \dot{x} &= A x + Bu \\ y &= C x + D u \end{align} $$ We know that the transfer function matrix $G(s) = C(s I - A)^{-1}B + D$ ...
0
votes
0answers
44 views

Efficiently solve a system of equations for only certain degrees of freedom given a known structure

I have an algorithm such that at some point I must solve the following system for $X_5$: $$ \left( \begin{array}{cccccccccc} A_1& B_1& & C_1& & & & & \\ B_7& A_2&...
-3
votes
1answer
35 views

Simplifying this matrix equation? [closed]

How can I simplify the following matrix equation? $$\frac{||A \times B||_{II}}{||A||_{II}}$$ where A, B are 3x1 matrices, <...
0
votes
1answer
22 views

Possible worlds/beliefs/Probability Matrix/Example 3

I the snippet below, copied from the Handbook of Game Theory with Economic Applications, the condition (2.1) says that we can rearrange the columns so that the matrix becomes block diagonal with each ...
0
votes
1answer
39 views

Difference between two Matrix representation of Linear Transformation

I have a linear transformation $$T(x_1,x_2,x_3) = (x_1-x_2+2x_3,2x_1+x_2,-x_1-2x_2+2x_3) $$ Assuming an ordered basis $\{u_1,u_2,u_3\}$ I get the linear transformation of the basis vectors as below: $...
0
votes
1answer
23 views

Proof for a property of graph laplacian and its complement?

Here it says Let $L(G)$ be the Laplacian of an undirected graph. $L(G)+L(G^c)=n I_n-J_n$ Where $J_n$ is a matrix with all entries $1$. Then how do we prove this (page $224$; eq. $5$): $$\...
0
votes
0answers
18 views

Solution of a set of (rank deficient) covariance matrix equations

I have a known covariance matrix $\Sigma_o \in R^{m \times m}$ and two known matrices $A \in R^{m \times n}$ and $B \in R^{m \times n}$ with $A \neq B$ and $m<n$. With that I want to compute a ...
0
votes
2answers
20 views

Gaussian Elimination with Partial Pivoting (What am I doing wrong?)

This should be a simple question Im using gaussian elimination with pivoting to solve \begin{bmatrix} 2 & 1 & 0 &| &3 \\ 1 & -1 &4 &|& -4 \\ 3 & -1 & -2 &...
0
votes
1answer
14 views

Signifiance of the Strictly Diagonallly Dominant matrix

Hello I am in a Numerical Analysis class and can't seem to find any information on this online or in the textbook. Strictly Diagonallly Dominant = SDD What is ...
1
vote
0answers
23 views

Matrix Equation $W(W^TW)=W(W_1^T W_1)$.

Let $W,W_1\in\mathbb{R}^{m\times d}$ matrices, with $m>d$, and $$ W(W^TW)=W(W_1^T W_1). $$ Suppose that $W_1$ is fixed. Can we characterize all $W'$s obeying the condition? For instance, if ${\rm ...
0
votes
0answers
26 views

Existence of a positive solution to a linear system Ax=b when b is not fixed

I'm interested in the existence of a vector $x \in \mathbb{R}^{n, +}$ such that for a matrix $M \in \mathbb{R}^{mxn} $ and a vector $b \in \mathbb{R}^m $, $Mx=b$. I've already looked into Farkas' ...
0
votes
0answers
29 views

How do you solve this matrix simultaneous equations?

This is from discrete systems analysis, you are given 2 matrix equations that you apply z transform to and then you need to solve for matrix Y. I have forgotten how to do these equations. Can someone ...
0
votes
1answer
37 views

Does the inner product of a matrix, $\frac{x^TAy}{x^Ty}$ stay the same or fall in a range for any x,y?

Is there any bound on $\frac{x^TAy}{x^Ty}$, for any vector $x$? I am observing that $\frac{x^TAy}{x^Ty}$ is approximately the same even when I change $x$. Why is that? Is there any property for the ...
0
votes
0answers
14 views

If $\frac{x^TAy}{x^Ty}\approx\frac{z^TAy}{z^Ty}$ then what can we say about x and z?

If $\frac{x^TAy}{x^Ty}\approx\frac{z^TAy}{z^Ty}\approx\frac{y^TAy}{y^Ty}$, where x is the eigenvector of A. Then what is the relationship between z and x? P.S.: In my case, $M=AB$ and $y$ is the ...
1
vote
2answers
55 views

Finding A solution to a matrix equation

I have the following problem in a past exam that involves matrix equations: Suppose that $X$is a $2 \times2$ matrix satisfying: $X^{2} = 6X +I$ a) Find the values of $\alpha$ and $\beta$ such that $...
0
votes
0answers
38 views

Exponential of Matrix Rewritten as Hyperbolic Functions

Working through a paper and cannot seem to confirm the following equality: $\exp(-\beta\hbar\omega_i\hat{S_{iz}}) = \cosh(\frac{\beta\hbar\omega_i}{2})-2\hat{S_{iz}}\sinh(\frac{\beta\hbar\omega_i}{2})...
-1
votes
1answer
64 views

Possible worlds/beliefs/Probability Matrix/Example 2

I have posted a similar question elsewhere, but this one is different. Take the definition and example as below as copied from the Handbook of Game theory with Economic applications. I want to design ...
0
votes
1answer
21 views

Gradient descent orthogonal steps

For the steepest descent algorithm it's stated that Since $\alpha_k$ minimizes $\alpha\mapsto f(x_k + \alpha p_k)$ it follows $$ \nabla f(x_k + \alpha_k p_k)^Tp_k=0. $$ where $p_k = -\nabla f(x_k)$....
0
votes
1answer
59 views

What is the meaning of a matrix eigenvalue to be 0? (Singular matrix, I know but what about it?)

$Av=\lambda v \implies$ $Av$ is parallel to $v$. If one eigenvalue is 0, then the determinant is 0 and the matrix is singular. What is the meaning of a matrix eigenvalue to be 0? Is it like $Av$ ...
1
vote
1answer
13 views

What is the meaning of “values coincides with the convex closure of the eigenvalues”?

In this paper, it says. What is the meaning of the sentence, "F(M) coincides with the convex closure of the eigenvalues if M is normal." I understand what is a normal matrix. But what does it ...
1
vote
2answers
89 views

Find $P$ where $P^{-1}AP$ for a given matrix $A$

I am doing a past paper and I have been given a matrix A: \begin{bmatrix} 4 & -1 & -3 & 2 \\ 4 & -2 & -4 & 4 \\ -4 & 4 & 6 & -4 \\ ...
0
votes
1answer
73 views

What do these matrices converge to?

Sorry this is quite a specific question, happy to reword the title however you think is more appropriate, but for a given $m$ x $n$ matrix A, and an initial, random $n$ x $k$ matrix $v$, I have this ...
0
votes
2answers
33 views

Under what conditions this equality holds?

Consider $P$ and $K$ be two $n\times m$ and $m\times n$ matrices. under which conditions $$ (I-KP)^{-1} K=K((I-PK)^{-1})$$ holds?
0
votes
0answers
15 views

How Field of Values changes the characteristics roots of a matrix?

Let $F(A) = x^*Ax$, $F(A')=x^*A'x$. If A has root $\lambda$ and A' has a root $2\lambda$, what is the relationship between $F(A)$ and $F(A')$? Does the field of values F(A) 'grows' as the ...
0
votes
0answers
31 views

Question about Baker–Campbell–Hausdorff Formula

Let $X$, $Y$, $V$ be matrices. $$e^{\tilde{V}}=e^{X}e^{V}e^{Y}$$ How to use the Baker–Campbell–Hausdorff Formula to prove the following identity? $$\tilde{V}=V+\frac{1}{2}[V,Y-X]+\frac{1}{2}\mathrm{...
2
votes
1answer
90 views

Help solving a tricky matrix PDE coupled system

I am trying to a solve a matrix equation of the form$$\begin{equation}\label{jacobi_cond} \frac{\partial \mathbf{P} (x_3)}{\partial x_3} = \mathbf{H}(x_3) \mathbf{P}(x_3). \end{equation}$$ where $$\...
-1
votes
4answers
65 views

If A and B are non-singular and n-square matrices. Show that $(I+BA)^{-1}=I-(((B^{-1})+A)^{-1})A$

A and B are non-singular n-square matrices. Show that $(I+BA)^{-1}=I-(((B^{-1})+A)^{-1})A$
0
votes
0answers
15 views

In Field of Values of a matrix, what is the meaning of $\lambda\epsilon \frac{F(A)}{F(B)}$?

Theorem 2 from this paper says, if $\lambda$ is an eigenvalue of $B^{-1}A$, then $\lambda$ $\epsilon $ $\frac{F(A)}{F(B)}$, where F(.) is the field of values. "F(M) is known to be a closed bounded ...
2
votes
2answers
77 views

Discrete-time LQR and solutions via LMI

Having a infinite horizon discrete-time LQR problem $J^* = \min_u \ \sum_{k=0}^{\infty} x^\top_k Q x_k +u^\top_kRu_k$ subject to $x_{k+1}= Ax_k+Bu_k, \quad x(0)=x_0$. With some algebra ...
-1
votes
1answer
43 views

Unknown 3x3 matrix, how to identify its identity matrix

An unknown 3x3 matrix A can be identified with the ero1, ero2, ero3, ero4. By following the order of operations below, it can be transformed into an identitymatrix: $ ero_1: \mathbf{r}_1 + \mathbf{r}...
1
vote
0answers
53 views

Bounds on $k=\frac{x^TBy}{x^Ty}$?

Let $\bf x, y$ be two normalized eigenvectors of two different matrices. $\bf{x}$ is a Fiedler vector, so perpendicular to $\mathbf{1} := [1,1,\dots,1]$. What can you say about the following? $$k := \...
3
votes
2answers
42 views

If $x = 2$ is a root of $\det\left[\begin{smallmatrix}x&-6&-1\\2&-3x&x-3\\-3&2x&x+2\end{smallmatrix}\right]=0$, find other two roots

If $x = 2$ is a root of equation $$ \begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3\\ -3 & 2x & x+2 \end{vmatrix} = 0 $$ Then find the other two roots. I solved it and got ...
0
votes
0answers
33 views

Given $u$ and $v$, looking for symmetric definite positive $B$ with $Bu=v$

I was wondering if $u$ and $v$ are such that $u^T v>0$, then there would exist an symmetric definite positive matrix $B$ such that $Bu=v$. Then to conclude that I need an orthogonal matrix $Q$ such ...