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

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32 views

Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
0
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3answers
40 views

Linear system $AX=0$ has a nonzero solution

Which parameters $a,b,c,d$ satisfy the matrix $$A=\begin{bmatrix}-1 & 1&1&1 \\1 & -1&1&1\\1 &1 &-1 &1\\a&b&c&d \end{bmatrix}$$ sucht that linear system ...
0
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2answers
21 views

Is there something called the Reduced Column echleon form?

I recently asked a question where I couldn't find the rank of a matrix. The question is : Problem on Finding the rank from a Matrix which has a variable At the time I believed in the answer, ...
0
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1answer
30 views

How do I rearrange this matrix equation to find X?

Given that the matrices $D$, $E$ and $F$ are invertible, how do I rearrange the equation to solve for $X$ when $D(X+3I)E = 5D(F+E) +E^2$. Would I just take the inverse of $D$ and $E$ to both sides ...
0
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1answer
8 views

Solution of system of linear equations.

Let $A$ be $n\times n$ matrix and there is at least one non trivial solution for the system $Ax=0$. For any real column vector $b$ with n components,the equation $Ax=b$ has: $1.$Unique solution. ...
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0answers
9 views

ODE -parabolic cylinder functions

How do we solve $\frac{d^2f}{dz^2} + \left(Az^2+Bz+C\right)f=0 \tag 1$ where $f(z),A,B,C$ are matrices of order $3 \times 3$.
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11 views

Determining matrix in terms of determinants of other matrices.

Determine |a+b e-f| |c+d g-h| in terms of the determinants of |a c| |b d| |a c| |b d| |e g| |e g| |h f| |h f| ...
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1answer
19 views

Skew symmetric Matrix - Commutative property

If A and B are two odd size skew symmetric matrices(for example $3 \times 3 $). Let us say $X=AB,Y=BA$ Question What is the general relationship between X and Y? Can we write Y using X?
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0answers
25 views

Find a basis for the four fundamental subspaces. [on hold]

Find a basis for the four fundamental subspaces of: $$A=\begin{bmatrix}1 & -1 & 0 & 2 \\ 0 & 0 &1 &1 \\ 0 &0 &0 &0\\0 &0 &0 &0\end{bmatrix}$$ I'm ...
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0answers
13 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
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1answer
15 views

How to prove $V*V^T=I$ in SVD? [duplicate]

How to prove $V*V^T=I$ in SVD: $M=U*S*V^T$? It's easy to understand $V^T*V=I$. It seems $V*V^T=I$, but how to prove it?
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2answers
24 views

What is the difference between a linear independent set and a generating set?

im having difficulty because onto have columns of generating set and one to one has columns of linear independence but the way we prove whether a standard matrix has linear independent columns are ...
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1answer
29 views

How to solve a matrix equation? [closed]

$$\pmatrix{1&5\\3&p}\pmatrix{w\\7}=\pmatrix{50\\35}.$$ How to solve this matrices equation?
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0answers
18 views

Solve the following system using Gauss-Jordan elimination

$4x - 8y = 12$ $3x - 6y = 9$ $-2x + 4y = -6$ So the augmented matrix will be: $$ \begin{bmatrix} 4 && -8 && 12\\ 3&& -6 && 9\\ -2 && 4 && -6 ...
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0answers
24 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...
1
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2answers
98 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
1
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0answers
22 views

Best approach to matrix representation of system of nonlinear ODEs

I have this system of ODEs: $$ \frac{dS}{dt}=\pi S-\beta S Z\\ \frac{dZ}{dt}=\alpha S Z - \delta Z $$ I am trying to rewrite them in the form : $$ \pmatrix{\dot{S}\\\dot{Z}}=\mbox{diag}(S,Z) ...
4
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1answer
46 views

Finding unknown matrices in a set of simultaneous matrix equations

I've come across a thorny problem in my research, which is too complicated and specific to ask here. However, it bears some similarity to the following problem, and understanding how to solve this ...
0
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1answer
39 views

Linear Algebra vs Matrix Algebra [closed]

Hi I don't know if this would be a proper question to ask here or not Anyway, I am an undergraduate electrical engineering student and I am considering taking another math course. What is the ...
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0answers
6 views

How to expand matrix representation of Ellipse

I try to study a classifier algorithm based on 2D gaussian distribution, and I find it hard to follow how the equation expands. According to a book, it defines a classifier with discriminate ...
2
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3answers
25 views

Solving for a matrix in equation form

Solve for $X$ assuming all matrices are n x n and invertible as needed. $$B(X+A)^{-1}=C$$ I solved this the following way: Multiply both sides by $(X+A)$ Multiply both sides by the inverse of $C$ ...
0
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2answers
27 views

Solving $CT = PC$ for transforms in $SE(3)$

I have three transforms: $C$, $T$, and $P$. Each of these transforms consists of 3D rotations and translations. I know $T$ and $P$, and I would like to solve for $C$. They are related by $T = C^{-1} P ...
3
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2answers
32 views

Eigenvalues of 3D rotation matrix

I'm having some trouble calculating the eigenvalues for this rotation matrix, I know that you subtract a $\lambda$ from each diagonal term and take the determinant and solve the equation for $\lambda$ ...
0
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4answers
84 views

If matrix $AB=A$, does it mean B must be an identity matrix?

If matrix multiplication $AB=A$, does it mean $B$ must be an identity matrix? If not, why? What conditions? $A$ is not a zero matrix.
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1answer
24 views

Convergence of Steepest Descent: Proving Orthogonality of Exact Line Search Steps

For the following assume that $f(x) = 0.5x^TQx - b^Tx$, where Q is symmetric, positive definite $n$ x $n$ matrix, and $b$ belong to $R^n$. Assume that $x^*$ is the unique local minimizer of $f(x)$ and ...
1
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0answers
77 views

Normal equations for minimization of Frobenius norm least squares error

I'm having a hard time understanding the most efficient sequence of steps for deriving the normal equations for Frobenius norm least squares minimization. Here I want to minimize the norm of a matrix ...
1
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1answer
13 views

Matrix norm inequality (p norm) proof

Given $I_{n} + A$ is nonsingular, prove: $||(I_{n} + A)^{-1}||_{p} \le \frac{1}{1-||A||_{p}}$ Now, I know that $I_{n} = (I_{n} + A)^{-1}(I_{n}+A)$, which simplifies to: $(I_{n} + A)^{-1} = I_{n} - ...
0
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1answer
37 views

Convert coordinates to a different coordinate axis

Sorry for any forum rules I have broken, I needed a quick answer. I want to create a plane including 3 nonlinear points on a 3d coordinate system, one being the origin. I also need to create a ...
0
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1answer
21 views

Write Generator Matrix (2,4) of Reed Muller code of (2,4)

I was wondering how to go about this sum, since I wasnt able to figure out how to solve this. Could any one help me out on this?
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2answers
48 views

Solutions of $Ax=b$ of square matrix $A$

If A is a $5 \times 5$ matrix and the equation $Ax = b$ is consistent for every b in $R^5$; is it possible that for some $b$, the equation $Ax = b$ has more than one solution? Why or why not?
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2answers
16 views

Matlab Code for substituting numbers in variable matrix

In Matlab, if we input a number matrix say [1 2; 3 4; 5 6], then what should i do so that the output would be of the form $ [(x_i-z)^b \hspace{5mm} y_i]$ where $z$ and $b$ are just variables. ...
0
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1answer
52 views

solve $X + X^T = A$

I have a simple question, for the following equation: $$X + X^T = A,$$ where $A$ is a symmetric matrix, can we solve $X$ from the equation? Thanks a lot. Feng [Edit] Sorry for this simple ...
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0answers
25 views

Solving N for HN=0, Given H is a special type of skew symmetric (n x n, n is a odd number) matrix.

Solving $N\ \mathrm{for}\ H \times N =0$, given $H$ is a special type of skew symmetric matrix $(n \times n, n\ \mathrm{is\ an\ odd\ number}\ n=2k+1)$, 0 on diagonal and 1, -1 in off-diagonal ...
0
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1answer
31 views

Show that a solution is linear and find its matrix in the usual basis.

On $\mathbb{R}^3$, define $$A(x) = (x \cdot a)a + 2(x \cdot b)b$$ Here $$a = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}, b = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ Show that A is linear and find ...
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0answers
23 views

Find an unknown vector in a matrix equation

Let's assume that $a$ is an $n\times1$ vector and $A_1$ and $A_2$ are $n\times n$ (not necessarily hermitian) matrix how can I find unknown vector $a$ in equation ${a^HA_1a}={a^HA_2a}$ ?
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26 views

how to derive this equation?

How can I derive this? $\min_{d_m} \|Y - DX\|_2^2 = \min_{d_m} x_m^Tx_md_m^Td_m - 2R_mx_m$ where $R_m = Y - \sum_{i \neq m } d_ix_i^T$ $x_m $ is a vector represents a row in $X$
0
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1answer
36 views

Solve an equation with an unknown vector

Let's assume that $a$ is an $n\times1$ vector and $A_1$ and $A_2$ are $n\times n$ hermitian matrix how can I find unknown vector $a$ in equation ${a^HA_1a}={a^HA_2a}$ ?
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0answers
41 views

Finding $A$ such that $R'=ARA^{T}$

Well i no this a silly question but i just want to know. There are two matrices $R'$ and $R$. We know that $R'$ is a diagonal matrix and we know $R$ and we also know that $R$ is a symmetric matrix. ...
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0answers
26 views

Matrix Inverse Question- Singular Matrix issue

I have a given Matrix equation $R(s)^{'}_{3\times 3} = \psi(s)_{3\times 3}R(s)\tag 1$ Conditions R(s) is orthogonal and determinent 1. Can say in the format of rotation matrix $R^{'}(s)$ ...
0
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0answers
17 views

solving for the trace and element-wise sum of matrix of Lyapunov equation

Assume the following continuous Lyapunov equation: $A\Theta + \Theta A^T = - D$ where $\Theta$ is the unknown target, and $D$ is a diagonal identity matrix. $A = - \textrm{Diag}(x^*)B$ and $x^* = ...
1
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0answers
20 views

Solve set of poorly conditioned linear equations in block matrix form

I would like to solve the following set of linear equations where A, B, C and D are each 4x4 matrices. K is then an 8x8 matrix The values in A and D have magnitudes of $\approx 10^{17}$, B has ...
0
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0answers
21 views

Solving the matrix equation $D P = D P A D$ for stochastic matrices.

Here, I call any real matrix with positive entries with rows summing to one a stochastic matrix (it need not be square). $D,A,P$ are stochastic. $P$ of size $n \times n$ is given. $D$ of size $k ...
2
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0answers
24 views

Skew symmetric matrices even size commutativity

Given Data in the question $w(t)=\frac{1}{2}\begin{bmatrix} 0 &r(t) &-q(t) &p(t) \\ -r(t)& 0 &p(t) &q(t) \\ q(t)& -p(t) &0 & r(t)\\ -p(t)&-q(t) ...
2
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2answers
54 views

Matrix Commutativity - Integration

Specifications and Data in question 1.We have a skew symmetric matrix $ A(t)_{3\times 3}= \begin{bmatrix}\,0&\!-a_3(t)&\,\,a_2(t)\\ ...
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1answer
49 views

sense of $\exp(x^\top A x) = \exp(A x x^\top)$

I wonder how the following identity makes sense? $\exp(x^\top A x) = \exp(A x x^\top)$ My approach: $\exp(x^\top A x) = \exp(\operatorname{tr}(x^\top A x)) = \exp(\operatorname{tr}(Axx^\top)) = ...
1
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1answer
31 views

Property of permutation matrices

I need the formal proof of this fact. I define the permutation matrix in this way given $\pi$ a permutation of $n$ elements its permutation matrix is: $${P}_\pi=\begin{bmatrix} {e}_{\pi_{1}} \\ ...
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1answer
14 views

Matrix problem: $W\beta=c$ and $F\beta=d$ implies $F=GW$ and $d=Gc$ for some $G$

Set up: Assuming we are working with real numbers, let $W$: $q\times K$ with $q<K$ and full row rank $c$: $q\times 1$ vector $F$: $K\times K$ matrix $\beta,d$: $K\times 1$ vectors Claim: ...
2
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1answer
44 views

Generalized eigenvalues of overdetermined systems

I have a system of equations that can be written as ${(\bf{A}} + \lambda{\bf{B}}){\bf{x}} = 0$ Where ${\bf{A}}$ and ${\bf{B}}$ are $n \times m$, integer matrices. I know that there are several ...
0
votes
1answer
46 views

How to solve the quadratic matrix equation

Given $\mathbf{A}$ and $\mathbf{B}$ two $m \times n$ real matrices, is there a closed form for the matrix equation \begin{equation} \|\mathbf{X}\|^{2}_{F} - 2\cdot trace(\mathbf{X}^T\mathbf{A}) ...
2
votes
3answers
159 views

Skew Symmetric Matrix Properties

We have a theorem says that "ODD-SIZED SKEW-SYMMETRIC MATRICES ARE SINGULAR" . Proof link is given here if needed. Now let us assume we have a $3\times 3$ skew symmetric matrices of the form $ ...