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

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5
votes
1answer
39 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
31 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
24 views

If positive definite matrices $A>B>0$ and $C>0$, then is $AC>BC$ true? [on hold]

Suppose I have positive definite matrices $A$, $B$ and $C$. If $A>B$, can we conclude $AC>BC$?
0
votes
0answers
4 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
39 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|||$.
0
votes
1answer
32 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 ...
1
vote
1answer
54 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
23 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
0answers
44 views

How to find one matrix, which is subject to $B^3 = A$. How much is such matrices? [duplicate]

Here I have a problem with row echelon form. $$A := \begin{bmatrix}-6 & 3 & 7 \\ 0 & -1 & 0 \\ -14 & 12 & 15\end{bmatrix}$$
0
votes
1answer
33 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
70 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 $|||⋅|||$ ...
0
votes
1answer
20 views

Gaussian elimination problem

$$x_1 + 10x_2 − 3x_3 = 8$$ $$x_1 + 10x_2 + 2x_3 = 13$$ $$x_1 + 4x_2 + 2x_3 = 7$$ when making 2nd and 3rd 1st columns 0 using Gaussian elimination, the second row second column also becomes zero, so ...
9
votes
3answers
184 views

If $\,A^3-A+I=0,\,$ then $A$ is invertible

Prove or disprove. If $A$ is a square matrix and $A^3-A+I=0,$ then $A$ is invertible. Is it possible to say the characteristic polynomial of $A$ is $\,p(t)=t^3-t+1$, and $A$ is invertible since ...
0
votes
1answer
31 views

Linear Algebra - Change of basis

Let $S$ be the standard basis for $\mathbb{R}^5$. Let $B = (b_1, b_2, b_3, b_4, b_5)$ be the ordered basis with: $b_1 = (2, 1, 1, -2, -2)$; $b_2 = (0, -2, 4, 5, -4)$; $b_3 = (1, -4, 5, 5, -4)$; ...
2
votes
2answers
49 views

Solution to $A = BX + YC$ where $A$ is a square matrix of rank $n$, $B,C$ known, rank $m<n$

I hope this isn't too trivial of a problem. I'm really struggling with it and I feel like it shouldn't be that difficult. As stated in the title: Given (full rank): $A\in\mathbb{R}_{nxn}$ ...
0
votes
1answer
19 views

How do you find the 4x4 matrix corresponding to the transformation T with respect to the basis?

If the transformation $T$ acting on the vector space $A \in Mat_{2,2}$ is given by $T(A)=CA$, where $ C= \left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right) $ how would you find the ...
0
votes
3answers
27 views

How to determine if the set of vectors are linearly dependent or independent

Determine if the following sets of vectors are linearly dependent or linearly independent $$V1=\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0\end{bmatrix}$$ $$V2=\begin{bmatrix}0 & 0 & ...
2
votes
0answers
27 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
-1
votes
0answers
14 views

Solve for X where Sum(A.X+C).(A.X + C) = X

I am trying to solve for X (vector of dimension n): $\Sigma(AX+C)\cdot(AX + C) = X$ where: C = is a constant vector dimension n A = is a diagonal invertible matrix [n,n]
0
votes
2answers
40 views

Why algebraic Lyapunov equation has an unique solution?

In the following text book (p.47): Optimal Control (Lewis 2nd edition) There is a theorem: (Zero input case) If $A$ is stable, and $(A,\sqrt Q )$ is observable, then $S_\infty= ...
0
votes
1answer
16 views

solving equation with linear span (using row reduction)

We've got the following span: $$U = Sp\{(2, 5, -4, -10), (1, 1, 1, 1), (1, 0,3,5) , (0,2,-4,-8)\}$$ We need to find the values of the number $a$ where the vector $$v = (a, a-6, 4a-3, 6a-1)$$ belongs ...
0
votes
0answers
25 views

How to get the unique generalized inverse matrix that we need?

For matrix equation Ax = b (A is a $3×4$ matrix, x is $4×1$ vector , b is $3×1$ vector). now, we have matrix A , vector b and already know that the third value of x is zero. How can we get the vector ...
0
votes
1answer
12 views

Can we ensure convergence for the jacobi method or do we simply trial and error?

For iterative methods for solving systems of equations, we may not always get convergence and it can depend simply on the way in which we write the equations. I understand there are tests which will ...
6
votes
2answers
107 views

Finding minimum from matrix

Consider following $3\times 3$ matrix. $\begin{pmatrix}3&6&9\\ 2& 4 &8\\ 1 &5& 7 \end{pmatrix}$ I need to find combination of three numbers where each number ...
0
votes
1answer
33 views

Solve matrix equation $AXB+CX=D$

How to solve matrix equation $AXB+CX=D$ for $X$? If it is not solvable, are there any numerical methods to do it?
1
vote
0answers
28 views

Formula for powering a matrix not working for all matrices

I'm currently learning about matrices and was asked to show that this formula works for powers of $M$. $$M^n = nM-(n-1)I$$ Where $M$ is the matrix (show below), $n$ is the exponent an $I$ is the ...
1
vote
0answers
21 views

How to find value of an unknown in matrix to make system of linear equations consistent

I'm currently stuck on this question relating to finding the unknown in a matrix so that the system of linear equations is consistent. I need to solve for $\lambda$. My first instinct is to try and ...
2
votes
2answers
94 views

Jordan matrix (de)composition

So any square matrix $A$ can be decomposed into $A = S J S^{-1}$ where $J$ has a normal Jordan form, moreover $A$ and $J$ are similar matrices. My question is quite straightforward. Given arbitrary ...
3
votes
0answers
31 views

Calculate the determinant of a matrix given a simple condition

I have been given this set of three inequations (involving three unknowns): $\underbrace{\left( \begin{array}{ccc} x-1 & -1 & -1 \\ -1 & y-1 & -1 \\ -1 & -1 & z-1 \end{array} ...
0
votes
0answers
17 views

System of linear equations with repeated equations

Suppose that I have this over-determined system of equations, $$a_1x_1 + a_2x_2 + a_3x_3 = k_1$$ $$b_1x_1 + b_2x_2 + b_3x_3 = k_2$$ $$c_1x_1 + c_2x_2 + c_3x_3 = k_3$$ $$d_1x_1 + d_2x_2 + d_3x_3 = ...
0
votes
1answer
44 views

Big trouble multiplying 2 matrix

I'm having a big trouble when I have to multiply 2 matrix. I think I have a problem with my calculator (HP 50g) because I get a correct answer but not the one my professor has. For example, I have to ...
0
votes
1answer
55 views

Reverse Order Laws of M-P pseudoinverse

When I was writing a literature survey on Moore-Penrose pseudoinverse (literatures like this one, and this one), I encountered with the following equality which was named as reverse order law: ...
0
votes
1answer
20 views

Find all the right inverses of a matrix

How do I find the right inverse of a non square matrix? The matrix i have is $$M = \begin{bmatrix} 1 & 1 & 0 \\ 2 & 3 & 1\\ \end{bmatrix}$$ Im really not sure how to even start this?
0
votes
1answer
21 views

Determine values of k for a matrix to have a unique solution

I have the following system and need to find for what values of k does the system have i) a unique solution ii) no solution and iii) an infinite number of solutions (k$^3$+3k)x + (k+5)y + (k+3)z = ...
2
votes
3answers
48 views

Using detA and detB to calculate the determinant of matrix C

If we have C=($A^t$)$^2$BA$^3$B$^-$$^1$A$^-$$^3$ and detA=-2 and detB doesnt equal 0, how do we calculate det C? I know that the transpose of a matrix does not affect the determinant. Does this mean ...
0
votes
2answers
38 views

How to show that $T$ is linearly independent?

This question came in my exam, and is by no mean a homework. Let $\{v_1,v_2\}$ be a linearly independent subset of a vector space $V$ and let $w$ belong to $V$ but not to ...
1
vote
1answer
33 views

Vandermonde determinant and linearly independent (corrected version)

This is a corrected version. Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 ...
2
votes
2answers
67 views

How to show that B is not invertible

This is not a homework, this question came in my exam, and I did not know how to solve it. "Let $A$ and $B$ be $3\times3$ matrices such that $AB = - BA$. Show that if $A$ is invertible, then $B$ is ...
0
votes
0answers
47 views

Is it possible to resolve equations of two vectors

I have a objective function as following $$F=\int |\alpha^TG(x)-w^TJ(x)|^2 H(x)\,dx+\lambda_1 \alpha^2+\lambda_2 w^2$$ where $\alpha^T$ is transpose of vector $\alpha= \begin{bmatrix} ...
0
votes
0answers
10 views

The z-axes of two orthoganal cartesian coordinates frames are aligned, then rotated about their x-axes by an angle. How do I calculate that angle?

I am trying to reverse a series of rotations applied to some Cartesian coordinate systems. Two coordinates systems, C1 and C2, are originally oriented with their z-axes aligned but not their x-axes ...
0
votes
0answers
23 views

solving a matrix quadratic equation

Suppose, I have the following expression: $x'(A-B)x = 0$ where $x$ is a vector that is unknown and $A$ and $B$ are matrices that are known. Both $A$ and $B$ can be assumed to be positive ...
1
vote
1answer
34 views

Lyapunov equation for stability analysis - what's the point?

Straight from Wikipedia: In the following theorem $A, P, Q \in \mathbb{R}^{n \times n}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite. Given ...
0
votes
1answer
58 views

Single factor model question, related to the benefits of diversifying one's portfolio.

The question: Suppose in a single period investment problem we may divide our wealth between n assets and that the return on the ith security is given by $r_i = \alpha + \beta_i\theta + \epsilon_i,$ ...
1
vote
1answer
13 views

quadratic form polynomial divisibility vs. matrix pointwise multiplication.

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
-1
votes
0answers
42 views

How to formulate the following problem as a SDP problem

I am struggling with the following optimization problem: maximize $\mu$, subjected to $\left(\begin{array}{ccc} 4\mu-(2\alpha_{10}+2\alpha_{5}+\alpha_{2}+\alpha_{8}) & 0 & 0\\ 0 & ...
0
votes
0answers
33 views

Using semidefinite programming to solve the following problem

I am struggling with the following problem, and wonder is SDP can help: $$\mathrm{maximize\ } \alpha_{10}+\alpha_5+(\alpha_2+\alpha_8)/2 \mathrm{\ subjected\ to\ } \mathrm{T_1}\succeq0, ...
0
votes
1answer
31 views

Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
1
vote
1answer
29 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
1
vote
1answer
29 views

How can I translate this problem into Matrix/Linear Algebra notation?

I have a matrix H of size s×d with holdings of s stocks across d days. H shows how many shares of each stock is in my portfolio on each day. The number of shares can change from day to day due to ...
1
vote
2answers
47 views

Finding a matrix from equation

we've got the following 4x4 Matrix $$\begin{pmatrix} 4 & -2 & 3 & 2\\ 3 & 5 & 1 & -4\\ -1 & 6 & -4 & -7\\ -2 & 0 & -2 & 4 \end{pmatrix}$$ and I need ...