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

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0answers
16 views

Solving eqn. of the form K = AGL + BGT, where A,B,L,T are invertible matrices.

I am obtaining the following equation in a regression problem: \begin{eqnarray} Z'_1Y_1\Omega^{-1}_{1}A+Z'_2Y_2\Omega^{-1}_{2}A = Z_{1}'Z_1\Pi A'\Omega^{-1}_1A + Z_{2}'Z_2\Pi A'\Omega^{-1}_2A ...
0
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1answer
22 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
0
votes
2answers
37 views

how to differential exponential of a matrix variable $f(X)=e^{X(t)\mathrm{d}t}$?

I have a function about a square matrix $X$ which depends also time: $f(X)=e^{X(t)}$, $t$ is time. So how to differential it about time to have $\frac{\mathrm{d}f(X)}{\mathrm{d}t}$? I know that ...
0
votes
0answers
25 views

What is the proof of this? (Matrices, Pivot)

I have a matrix : $A$ I pivoted $A$ with a pivot element $(p)$ and I get this matrix: $B$ What is the proof of this equation? $|A|$ = $\frac{1}{p}. |B|$
1
vote
1answer
21 views

Transforming a square matrix A into B

Let's say I have $A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}$ and $B= \begin{bmatrix} b_{11} & ...
0
votes
2answers
15 views

Solving systems of equations using matrices by row reduction

Solve the following system for $a$, $b$, and $c$: $$\begin{pmatrix}1 & -1 & 2\\2 & -2 & 2 \\ 3 & -3 & 2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = ...
3
votes
2answers
63 views

The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
0
votes
2answers
49 views

Solution to a system of linear equations with an unknown matrix product

Consider the system of equations $$ Xy=Ab $$ where $X$ and $A$ are $m \times m$ invertible matrices and $y$ and $b$ are $m \times n$ matrices. The matrices $X$ and $y$ are unknown and the matrices $A$ ...
3
votes
2answers
68 views

Problem 11 Section 2.6 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Let $X$ be the vector space of all complex $n \times n$ matrices and define $T \colon X \to X$ by $Tx \colon= bx$, where $b \in X$ is fixed and $bx$ denotes the usual product of matrices. I know that ...
1
vote
2answers
62 views

When $AX=BX \Rightarrow A = B$?

Given the following matrices equation: $AX=BX$ under which assumption we can say that $A=B$? The obvious one is when $X$ is invertible. Is there any other ?
1
vote
3answers
69 views

How to get matrix $A$ from $A^\top A=B$ with given symmetric matrix $b$?

Given a symmetric matrix $B \in \mathbb{C}^{n\times n}$. How many coefficients of $A \in \mathbb{C}^{n\times n}$ can you obtain from the following equation? $$A^\top A=B$$ I think this problem is ...
0
votes
1answer
22 views

Solving the linear system of equations

Is it possible to solve non square matrix with Gaussian eliminations? OR any other way to solve a 6 equation with 8 variables?
1
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1answer
37 views

If the null space contains only the zero vector, the map is one-to-one

How does finding out if the null space has only the zero vector prove one-to-one? One-to-one means that there are distinct images for each distinct vector input. $$\mathbb R^n \to \mathbb R^m$$ ...
1
vote
1answer
49 views

What is the correct $\det(A^{-1})$

Ok so I think I know why this is incorrect, because of the following: $$\det\frac{1}{ad-bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}\neq \frac{ad-bc}{ad-bc}$$ However, by adding a det ...
2
votes
1answer
51 views

Solving equation with integer matrices as unknowns

I am currently working on a problem, where I need to know for square integer matrices, $A$ and $B$, whether or not there exists square integer matrices, $X$ and $Y$, such that $X(A-I)Y=B-I$, where $I$ ...
0
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1answer
23 views

Solve Unknown Matrix Variables

I have a markov chain matrix with probabilities as such, on finding the steady state.. ...
2
votes
0answers
36 views

Number of solutions of a matrix algebraic equation

Reading that, I found that: given a matrix algebraic equation $$ X^n+A_1X^{n-1}+\cdots + A_n=0 $$ where the cofficients $A_1\cdots,A_n$ as well as solutions $X$ are supposed to be square complex ...
0
votes
2answers
46 views

How to get the perfect square for the following equation

The problem is defined as follows: $$ \min_X tr(X^T A X)-\alpha tr(X^T B) $$ I want to get the equal perfect square equation as that above, that is $$ \min_X \| X-C\|_F^2 $$ where $C$ is related to ...
-1
votes
1answer
27 views

5 x 5 eigenvalues question

Let $λ_1, λ_2,λ_3,λ_4, λ_5$ be the eigenvalues of the matrix $A$, where $A$ is given by: $$ \pmatrix{0&0&1&0&0 \cr 0&1&0&0&0 \cr ...
1
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2answers
46 views

Linear Algebra - Proof that the linear system has a solution.

I have this problem : This is a linear system with two equations, and four unknown. $$a_1x_1+a_2x_2+a_3x_3+a_4x_4=a$$ $$b_1x_1+b_2x_2+b_3x_3+b_4x_4=b$$ These vectors ...
0
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0answers
26 views

Trying to prove any symmetrical matrices would be a vector space

Here is the question I am struggling with; Let $V$ be the set of any real symmetric matrices, that is, the set of all matrices $A$ such that $A^{T}=A$ For whatever reason I just can't seem to find ...
5
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2answers
95 views

Consider $AX + XA = B$, how many equations and how many unknowns?

Here, $A$, $X$, and $B$ are real $n\times n$ matrices. A and B are given, but X is unknown. Also, A is symmetric and positive definite, so $A^T = A$ and $z^TAz >0$ for all nonzero $z$ in $\mathbb ...
0
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2answers
22 views

True/false questions on image kernel and basis of vector spaces and subspaces.

1) The set ${t + 1, t2 + 2, t2 + t}$ is a basis of $F_3[t]≤2$. I put false because if t is 2, then we have ${t + 1, 0 , t2 + t}$ so a non zero coefficient could exist. 2) T : V → V a linear ...
2
votes
2answers
50 views

Prove that $AB-BA = I$ has no solution in $M_{n\times n}(\mathbb R)$ without using matrix trace

The title is self-explanatory. Prove that $AB-BA = I$ has no solution in $M_{n\times n}(\mathbb R)$ without using matrix trace. A,B are both from $M_{n\times n}(\mathbb R)$ and $AB$ is matrix ...
3
votes
1answer
21 views

Find the elements of a matrix

I know this is a very simple example, and I guess the solution is something very easy, but I just can't quite understand what exactly should be done in the task and what approach should I take: Is ...
0
votes
0answers
16 views

transformation of a point with constrained movement between two frames

Suppose I have two reference frames (3D). Frame F1 is the static world frame, ie. stationary and with no rotation. Frame F2 is a freely moving frame, but always stays on the positive Z side of F1. ...
0
votes
1answer
28 views

How to calculate this matrix in component-form? (Undergrad)

If ${A}_{ab} = \delta_{ab} + \varepsilon_{abc}n^c$ and $B^{ab} = \frac{1}{1+n^2}(\delta^{ab} + n^an^b - \varepsilon^{abc}n_c)$ what is the correct way to evaluate $$C^{ab} = (AB)^{ab} $$ Here, ...
0
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0answers
16 views

On Farkas's Lemma and Existence of a particular solution

This is a real life problem. I have a matrix $A$ which is $m\times n$. I want to check for the conditions on the existence a vector $x\in\mathbb{R}^n$ such that $A x \geq 0$. The Farkas's Lemma, as I ...
0
votes
1answer
26 views

How to merge similar terms to get a perfect square form?

There is a objective function that has the following form: $$ \alpha \|X^T AX\|_F^2-trace(B^T X) +\beta\|X-C\|_F^2 $$ where $\alpha,\beta$ are scalars, and $X,A,B,C$ are compatible matrices. ...
0
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0answers
13 views

Using the inverse of the matrix find all the solutions of the following systems of equations?

I found the inverse using row operations and the identity matrix but I dont know where to go from here. Can someone direct me please ?
0
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0answers
19 views

Find solution of signle element $y_i$ in vector $y$ subject to $Ay=c$

I have a interesting question about linear algebra problem. Assume that I have a matrix $A^{m \times n}$ and vector $c^{n \times 1}$ are known and I want to find the solution of vector y subject to ...
1
vote
5answers
126 views

If $A^n=0$, then $I_n-A$ is invertible. [closed]

How do I solve this problem? $A$ is $n\times n$ and $A^n=0$. Prove that $I_n-A$ is invertible.
0
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1answer
28 views

Resolve $Ay=b$ with fast method

I am looking for method that resolve equation $$Ay=b$$ I read the paper "Wiedemann's algorithm" that is one solution for fast way to find the solution instead of Gauss-Elimination. Could you suggest ...
0
votes
1answer
26 views

Showing that the eigenvectors (when eigenvalue is 1) can be chosen to be integer valued

Suppose $A$ is an $d \times d$ matrix with integer entries. If there exists $\underline{n} \neq 0 \in \mathbb{R}^d$ such that $(A^T)^k \underline{n}= \underline{n}$. How can you show/justify that ...
0
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0answers
7 views

Matrix transfrom $AXB=P$ to $A'X=P'$

I have a series of linear equations: $P = aS^0+bS^1+cS^2+dS^3$ $a = a_0R^0+a_1R^1+a_2R^2+a_3S^3$ $b = b_0R^0+b_1R^1+b_2R^2+b_3S^3$ $c = c_0R^0+c_1R^1+c_2R^2+c_3S^3$ $d = ...
0
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0answers
32 views

Can anyone help me deduce a matrix inequality?

The following lemma is taken from references firstly. Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying ...
0
votes
2answers
27 views

$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?

$A,B,X,Y$ are four invertible matrices. If $AYB=XY$, can I express matrix $X$ in the terms of $A$ and $B$?
0
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1answer
15 views

working with matrices, mean solution?

I don't understand why the author says that following matrix equation solution, is a "mean solution" see, find out $[R]$ $$[N][R] = [B]$$ where $[N]$ is a 3x4 matrix, $[R]$ is a 1x3 matrix and ...
2
votes
2answers
31 views

Use the cayley hamilton theorum to work out high powers of matrices

Let Matrix $$A= \left( \begin{array}{ccc} 1 & 2& 3 \\ 0 & 1 & 0 \\ 0 & 5 & -1 \end{array} \right) $$ Compute $A^{25}$ using the cayley hamilton theorum I know i use ...
1
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0answers
19 views

How to rewrite this matrix form

I had this equation. \begin{equation} \begin{pmatrix} g_{1,1} & g_{1,2} & \cdots & g_{1,n} \\ g_{2,1} & g_{2,2} & \cdots & g_{2,n} \\ \vdots & \vdots & \ddots ...
0
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1answer
22 views

Composition of binary quadratic forms as matrix operations

It is easy to see that any binary quadratic form $a^2 + 2bxy + cy^2$ is the same as $XAX^T$ where $X = [x, y]$ and $A = \begin{bmatrix}a & b\\b & c\end{bmatrix}.$ The composition of two ...
3
votes
3answers
50 views

Proof of transpose property of matrix exponential

Using the fact that the matrix transpose distributes over infinite sums to show that $e^{(A^T)} = (e^A)^T$. I feel like this is really trivial, but I don't know quite how to prove this. How would I ...
0
votes
0answers
21 views

Finding subspace

Determine subspace's base $W\subset \mathbb{R}^5$ desribed by the equation AX=0 where $A = \left( \begin{smallmatrix} 2&1&2&1&3\\ 1&1&-1&2&1 \\ ...
2
votes
1answer
33 views

Checking if a set of vectors are a basis

Find a basis for the subspace \begin{align} V=\{ (x_1,x_2,x_3,x_4)^T\in \mathbb{R}^4 | x_1-3x_2+5x_3 -6x_4=0\} \end{align} where $V\subset \mathbb{R}^4$ The basis ends up being spanned by the ...
0
votes
0answers
37 views

How to make a complicated equation more readable?

I have a very complicated equation: $y=(((g_0'(1) b_1-g_0'(1)) b_2-g_0'(1) b_1+g_0'(1)) \gamma^2+((g_0'(1) b_1-g_0'(1)) b_2 n+((g_0''(1)-4 g_0'(1)) b_1+3 g_0'(1)) b_2+(g_0'(1)-g_0''(1)) b_1) ...
0
votes
1answer
14 views

Confused about the dimension of a span of a set of vectors ls

The question is: What is the dimension of the following subspace of $\mathbb{R^5}$? $$span\left( \begin{pmatrix} 1 \\ 0 \\ -1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ -1 \\ 1 \\ 0 ...
0
votes
0answers
21 views

Solution for set of matrix equations involving an inverse

I am encountering the following set of three matrix equations for which I search a solution in terms of ${\bf M}\in\mathbb{R}^{N\times N}$ and ${\bf D}\in\mathbb{R}^{Q\times N}$, $${\bf M}{\bf W} = ...
2
votes
0answers
23 views

Matrix equation system with orthogonal matrices

$T_1,T_2$ are unknown orthogonal matrices $3\times3$ with determinant 1, $v_i,u_i$ are known vectors $3\times1$. How can i elegantly solve this ? $T_1v_1=T_2u_1$ $T_1v_2=u_2$ $T_2v_3=u_3$
1
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0answers
39 views

Solution of equations involving determinant and matrix inverse

$x$ and $y$ are two scalar unknowns. The two equations are $$|\mathbf{I}+x\mathbf{h}_1\mathbf{h}'_1+y\mathbf{h}_2\mathbf{h}'_2|=R$$ and ...
3
votes
3answers
257 views

For which $x$ is the determinant vanishes?

For which values of $x \in \mathbb{R}$ does the determinant of the matrix $$ M = \begin{pmatrix} x & 0 & 1 & 2 \\ 2 & x & 0 & 1 \\ 1 & 2 & x & 0 \\ 0 & 1 ...