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

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0answers
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Solving a matrix equation problem

I've been trying to solve this matrix equation but I just can't find the way to do it. The task is: Find matrix $\mathbf{X}$ from the equation in the photo where $\mathbf{A, B, C}$ are given matrices ...
3
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0answers
42 views

Result about Matrices of form $B(AB)^{-1}A$

I am trying to prove the following result. So far my only idea was to try using the formula for inversion of block matrices, but that did not get me very far. Any help will be much appreciated. ...
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0answers
10 views

Question with Gauss-Jordan elimination produces the matrix

Please help me explain the problem below: How can I use Gauss-Jordan to get all bottom roll become 0? Thank you so much!
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0answers
20 views

How to compute the gradient of the following matrix function? [on hold]

$f(X)=\left\|XX^T-I \right\|_F^2$, where $\left\|\cdot \right\|_F^2$ is Frobenius matrix norm
0
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2answers
22 views

solving an XOR matrix

I'm working on a somewhat-unique linear algebra problem arising from XORing files together in order to encode them, and then figuring out how to subsequently recreate the original files from the ...
0
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1answer
18 views

Consistency of system of linear equations without taking it to echelon form

Establish the conditions under which the equations $$ax + by + cz = q-r;bx + cy + az = r-p;cx + ay + bz = p-q ,$$ are consistent. I am aware that by taking the system to echelon can get me the rank ...
0
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0answers
11 views

Commutivity with Jordan Decomposition of Linear Transformation

Let $T \in L(V,V) $ and $T = D + N$ Where $D + N$ is the jordan decomposition of $T$ and $D$ is diagonable Let $X \in L(V,V)$ Show $XT = TX$ iff $XD = DX$ and $XN = NX$ I started out with saying ...
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1answer
18 views

Why some of the value in inverse matrix become positive?

Ans: But why? Isn't it calculate something like this: Please help explain. Thank you!
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1answer
18 views

how do you solve the simultaneous equations 2x + y = 9 and x - 2y = -8 using the matrix method?

i first convert the bases into a 2x2 matrix and then i multiplied the inverse matrix by the 2x1 e/f matrix of 9 (on top) and -8 (on the bottom) which gave me x = 5.4 and y = 5 however the answer ...
1
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0answers
17 views

Is it true that $\|(I+A)^{-1}\| \leq \|(I-B)^{-1}\|$ if $|A|<B$?

Is it true that $\|(I+A)^{-1}\|_\infty \leq \|(I-B)^{-1}\|_\infty$ if $|A|<B$? What if $A$ and $B$ are triangular matrices? Definition 1: $|A| = [|a_{ij}|]$, absolute value of every entry of $A$. ...
1
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1answer
37 views

Various matrix manipulations effect on determinant

Suppose that the matrix $A$ below has determinant $−3$. Find the determinants of $B$, $C$ and $D$. $$ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} ...
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0answers
33 views

Whether $\left| {{x^T}Ax} \right| \le \rho (A){x^T}x$ holds for an arbitrary invertible matrix $A$? [closed]

where $\rho(A)$ is the spectral radius of $A$, $x$ is an arbitrary column vector.
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4answers
36 views

Find the limit a matrix raised to $n$ when $n$ goes to infinity

Let $ A $ be a $ 3\times3 $ matrix such that $$A \left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right)=\left( \begin{array}{ccc} 1 \\ 2 \\ 1 \end{array} \right),~~~A \left( \begin{array}{ccc} ...
1
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2answers
42 views

$n\cdot tr(AB)=tr(A) \cdot tr(B) $ $A$ is a scalar matrix

Let $A\in M^{n\times n}(\mathbb R)$. prove that if for every other $B\in M^{n\times n}(\mathbb R)$: $n\cdot tr(AB)=tr(A) \cdot tr(B) $, $A$ is a scalar matrix.
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1answer
30 views

Prove Or Disprove: $n\cdot tr(AB)=tr(A) \cdot tr(B) $iff $A$ or $B$ is scalar matrix [closed]

Prove/Disprove: $n\cdot tr(AB)=tr(A) \cdot tr(B) $ iff $A$ or $B$ is scalar matrix. A and B are square matrices of size n. So far I managed to prove that one side is right, left to right, that if A, ...
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0answers
21 views

Construction of matrices with special property

Can any one help me in constructing two matrices $A$ and $B$ (can be singular) such that $R(A)\subseteq R(B)$ and $R(B^*)\subseteq R(A^*)$. Here $A^*$ means conjugate transpose of $A$. It will be ...
2
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1answer
22 views

Solve matrix solution $TUT^{-1} = t^{m}U.$

Let U be nilpotent nxn matrix. $U^{n} = 0, U^{k} \neq 0$ if $k<n$. Therefore, in some basis U is Jordan block, i.e. $U = J_{0,n}$. Let T be semisimple matrix that in some basis is $\begin{pmatrix} ...
0
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3answers
24 views

Prove that a matrix is a scalar matrix

Given $k\in\mathbb{N}$, an invertible matrix $M$, and the equation: $(M^{-1}AM)^k=3I$ I need to prove $A$ is a scalar matrix, without using eigenvalues. I understand why it's true, but can't ...
0
votes
1answer
20 views

Using Cramer's Method

How do I use Cramer's method to solve the following system of equations ? 2x+2=10 2y=2 2-3y=6x I've solved for y using standard simultaneous equations but this didn't help
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1answer
29 views

Rewrite expression $b^Ty=a^Tx+C$

A really simple question. For four $n$-dimensional vextors $a,b,x,y \in \mathbb{R}^n$ and a scalar $C \in \mathbb{R}$, it is known that $b^Ty=a^Tx+C$. Having another vector $d \in \mathbb{R}^n$, how ...
0
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2answers
11 views

If spectral radius $\rho(A)<1$ , does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-||A||_{2})$ hold true?

If spectral radius $\rho(A)<1$, does the inequality $||(I-A)^{-1}||_{2} \leq 1/(1-\||A||_{2})$ hold true? If it is correct can somebody give me link to the proof for this inequality?
1
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1answer
51 views

Differentiation of determinant. Is $\frac{d}{ds} L = LW$ implies $\frac{d}{ds} \det(L) = \det(L) tr(W)$?

For a matrix $L$ and $W$, Is $\frac{d}{ds} L = LW$ implies $\frac{d}{ds} \det(L) = \det(L) tr(W)$? Is $\frac{d}{ds} \det(L) = \det(\frac{d}{ds}L)$ true? ( I guess not)
3
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0answers
40 views

Convergence of iteration scheme of solving matrix equations

Consider the equation $A\mathbf{x}=\mathbf{b}$. It is equivalent to $$S\mathbf{x}=(S-A)\mathbf{x}+\mathbf{b}$$ where $S$ is a splitting matrix. We now consider the iteration scheme ...
0
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0answers
11 views

Constructing a formula

I have two matrices A,B each item in A is tested against all items in B. If (a,b) matches then I have to calculate a weight for ...
0
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0answers
11 views

generalised eigenvalue problem with absolute value

Problem: $\max_w |w^t A w|-|w^t B w|$ s.t $w^t C w=1$ If there was no absolute values, i.e. if the problem was $\max_w w^t A w-w^t B w$ s.t $w^t C w=1$ this would, by using the appropriate Lagrange ...
0
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0answers
46 views

A question about the “state-transition-matrix” of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
1
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1answer
79 views

Computing a matrix from its exponential

Given the exponential of a matrix, $e^{At}$, what is the best way to compute A? Thinking about complex-valued 3x3 matrices here.
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1answer
32 views

How do I reasoning matrix of size n to get the big-o?

The following very simple algorithm calculates the determinant of a matrix in a recrusive fashion with no optimization at all. Count how many operations (+, -, * and /) are done for a matrix of size n ...
5
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1answer
83 views

Prove that trace of a matrix is $0$.

Let $ n\geq 2 $ and $ A,B,C \in M_{n}(\mathbb{C}) $ be three matrices so that $$ A^{2}B+BA^{2}=2ABA $$ and $ C=AB-BA $. Prove that $ \mbox{tr}(C^{k})=0,\forall k\in \mathbb{N}. $ I tried solving it ...
1
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1answer
25 views

Express $\frac{d^3y}{dx^3}-\frac{dy}{dx}+y=\cos(x)$ in matrix form $x'=Ax+f$

As the title says, I need to write $\frac{d^3y}{dx^3}-\frac{dy}{dx}+y=\cos(x)$ in the following matrix form $x'=Ax+f$ I've rearranged the equation to be in that form, but how do I extract my ...
0
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0answers
17 views

Propagation of Errors, Variance of a $N$ dimensional vector

I am working on a calculation and I'm trying to see if the last step is true. I tried to simplify all the details of the problem. Let $\mathbf{W}$ be a $N\times N$ matrix, $a=\{a_{1,}a_{2},a_{3}\}$ ...
0
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0answers
56 views

What does multiplying a matrix A by B and B transpose mean?

$$BA^TB^T$$ $A$ is ($m\times m$) while $B$ is ($n\times m$). In matlab this is written as: B*(A\B'). I don't understand what it means for $A$. What kind of transformation does it do on $A$? ...
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0answers
12 views

Given hermitian $C$ and diagonal $\Delta$, find $C=H\Delta H^H$ with $H$ orthogonal.

Given $C$ a hermitian complex non-singular square matrix, and $\Delta$ real and diagonal with distinct, non-zero diagonal elements, I need to find an orthogonal matrix $H$ such that $$C=H\Delta H^H.$$ ...
0
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0answers
13 views

Converting homogeneous projection matrix

I have a 4x4 homogeneous projection matrix which converts 3D world space coordinates into 2D image coordinates + a depth value. It is of the form $\mathbf{H} = \begin{bmatrix} m_{1,1} & ...
0
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0answers
47 views

Question about recurrence relation using vectors and matrices.

I have an interest whether the following recurrence relation can be solved. My Problem: $$ \mathbf{x}_{k+1}=\alpha \ \mathbf{x}_{k}+\biggl[ \mathbf{E}-\alpha \ \biggl(\cfrac{\mathbf{x}_{k} \ ...
0
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0answers
33 views

Can this matrix equation be solved?

I have a matrix equation of the form [Z][C][Z] - [Z][D] = [A][Z] - [B] and I need to solve for [Z]. Is it possible? If so, what is the solution? The matrices are all general, complex, with no ...
4
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2answers
245 views

Finding a $5\times5$ Matrix such that the sum of it and its inverse is a $5\times 5$ matrix with each entry $1$.

How can I do this question: Is it possible to find a $5 \times5$ invertible matrix $B$ over $\mathbb Z_2$ such that: $B+B^{-1}=5\times5$ matrix where every element is $1$ anyway, I assume that the ...
0
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1answer
37 views

Matrix Algebra!

If we have a matrix multiplication equation, $$s=A^\top \cdot P \cdot B,$$ Where $s$ is an integer and I have to find the matrix $P$. How would I use matrix alegbra here, since $A$ and $B$ are ...
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2answers
38 views

Solve matrices algabraic

The problem is given below: In the following $X, A, B$ and $C$ are all $n \times n$ matrices. Solve for $X$ and account for any assumptions made for each of the following. (a) $AX + B = C$ ...
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1answer
31 views

matrix with two unknowns

I am to calculate the value of this matrix $$ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & a \\ 1 & b & 1 \end{bmatrix} $$ I do a basic ...
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0answers
27 views

Matrices and cofactors question

From Mathematical Methods for Engineers and Scientists 1 by K.T. Tang: http://imgur.com/A99uGc5 Why did the author do part b differently than part a (aside from the obvious fact the the problem ...
0
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1answer
17 views

Values for $*$ symbol in linear algebra equation

In the following equation shown below, I am wondering as to how to interpret the $*$ symbol. What values of $j$ in $p_{i*}$ should be used, and what values of $i$ should be used in $p_{*j}$ ...
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2answers
57 views

All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar

If $A^2 + I = 0$ is a matrix equation, all solutions $A \in M(n,\mathbb{C})$ are similar to $$ B = \begin{pmatrix} i I_p & 0 \\ 0 & -i I_m \end{pmatrix} $$ where $i$ is the imaginary ...
3
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0answers
54 views

$A,B \in M_n(\mathbb{R})$ are similar over $\mathbb{R}$ if they are similar over $\mathbb{C}$

I want to prove the following statement: Assume that $A,B \in M_n(\mathbb{R})$ and there is an invertible matrix $P \in M_n(\mathbb{C})$ that $PAP^{-1}=B$. Prove that there is an invertible ...
0
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2answers
67 views

In Ax=b. If A is not invertible there are no solutions or infinity. How to determine what the case is?

I learnt that for the equation: $Ax=b$ There is one solution if A is invertible. But if $A$ is singular there are infinity solutions or no solutions at all. If $A$ is singular is it possible to ...
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1answer
33 views

How to express the 'general solution' of a matrix with a unique solution [closed]

Hi guys so I've been given some uni homework for the subject Calculus and Linear Algebra. Basically it's just your standard 'give the general solution of the system, or show that no solutions ...
2
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0answers
30 views

LU Decomposition vs. QR Decomposition for similar problems

Suppose I want to solve the 2D Poisson equation with Neumann boundary conditions. The solution is non-unique up to an additive constant. I have previously asked a related question here for the 1D ...
0
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0answers
28 views

Is my proof correct? (prequel to inverse matrices)

This question comes from a section before inverse matrices are introduced. Suppose $AD=I_m$. Show that for any b in $R^m$, the equation $A$x$=$b has a solution. [Hint: Think about the equation ...
0
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1answer
18 views

Rearranging a Matrix equation

Can someone please help me rearrange this matrix expression: $$T=S(F^T)^{-1}=JF^{-1}\Sigma(F^T)^{-1}$$ I need $\Sigma$ in terms of all the other variables. I understand what to do when it is ...
2
votes
1answer
32 views

For invertible $A$, $C$, prove that: $(A^{−1} + B^TC^{−1}B)^{−1}B^TC^{−1} = AB^T(BAB^T + C)^{−1}$

Here's what I have to prove: For invertible $A$, $C$, prove that: $$ (A^{−1} + B^TC^{−1}B)^{−1}B^TC^{−1} = AB^T(BAB^T + C)^{−1}. $$ Here's what I have so far. I couldn't do the proof entirely ...