For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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73 views

Determine over $\mathbb{Q} $ and $\mathbb{C}$ all $3 \times 3$ matrices with $A^4 = I$

I know that if $A$ is a $3 \times 3$ matrix whose minimal polynomial divides $x^4 -1$, then the minimal polynomial has the restriction that has at most degree $3$. On the other hand, we have: $$x^4 ...
1
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1answer
20 views

Continuity on kronecker product

Let $\{A_k\}_{k\in{\mathbb{N}}}$ a sequence of $n\times n$ matrices. Suppose that $A_k \to A$, and consider $B$ another $n\times n$ matrix. Its true that... $A_k \otimes B \to A \otimes B$ ?
0
votes
1answer
32 views

How do you find the subset of $2\times3$ matrices that forms a basis for a subspace $V$?

I know know how to find a seubet of vectors that forms a basis but how can you from a basis with matrices?
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0answers
23 views

Prove that a transformation is linear [closed]

Prove that if $M$ is an $m \times n$ matrix then the transformation $T:{\mathbb{R}^n} \to {\mathbb{R}^m}$ given by $x \mapsto T(x) = M \cdot x$ for any $x$ that exists in ${\mathbb{R}^n}$ is linear. ...
1
vote
0answers
18 views

Creating a matrix so its Euclidean norm (p=2) is < 1

I have to convert $Ax=b$ to the form $x=Bx+b$, so $\|B\|_2 < 1$. I'm having serious trouble with picking/creating the matrix $B$ and it's making me nuts. Can someone give me an example or a hint? ...
0
votes
0answers
14 views

4 D rotation matrix

I have 2 vectors with 4 elements each and they are perpendicular to each other. z = [ -0.0310 -0.0894 -0.9451 -0.3128] and w = [0.9451 0.3128 -0.0310 -0.0894] how do i compute the 4x4 rotation ...
1
vote
1answer
48 views

A matrix similar to a basis change matrix is also a basis change matrix [closed]

Please tell me whether this statement is right or not. I think it's right but I can't prove it. A matrix similar to a basis change matrix is also a basis change matrix.
1
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1answer
27 views

How should I build a SU(4) matirx with a C4 vector?

I have a complex vector $S=[S_1,S_2,S_3,S_4]$ with $|S_1|^2+|S_2|^2+|S_3|^2+|S_4|^2=1$. My question is how to bulid a matix $C\in SU(4)$ while \begin{equation}C= \left( \begin{array}{cccccc} S_1 ...
0
votes
1answer
29 views

Need help with a better understanding of change of basis matrix and corresponding theorems

I'll try to summarize here what I understand so far about the concepts of change of basis matrix etc. Let $\beta$ and $\gamma$ be two different ordered bases for the vectorspace $V$, and let $P ...
1
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1answer
28 views

Bound on the eigenvalues of PSD matrix [duplicate]

Given That A and B are two PSD (positive semi-definite) real matrices and the following holds $$ A \leq B $$ (meaning that $$ B-A $$ is also PSD) can I bound the eigenvalues of A using the ...
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0answers
12 views

$Mod 3$ or $mod 5$ for for an $N5(3, 5)$ is a $[5, 2, 4]5$ Reed Solomon code.

I am trying to reduce a matrix to the form $G = [I|-B^T]$ but I just cannot get my solution to match the solution in my notes. I am reducing it Mod 3 as it is part of a coding theory question in ...
0
votes
2answers
54 views

Counting Distinct matrices

How many distinct (if matrix $M$ is included in count, do not include $PM$ where $P$ is permutation matrix) $3\times 3$ matrices with entries in $\{0,1\}$ are there such that each row is non-zero, ...
0
votes
0answers
26 views

Breadth First Search Question

The adjacency matrix for a simple graph is shown here (as an HTML table). The questions below refer to the graph that this table represents. Give the path that is found from vertex G to vertex E ...
0
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1answer
12 views

Example of a matrix which is copositive plus but not PSD.

This came up in our game theory course. While doing the Lemke's algorithm for solving LP, it was said that the process terminates when the matrix $M$ is copositive plus. Now copositive plus has a ...
0
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0answers
22 views

Problem reducing a matrix modular 3

I am trying to reduce a matrix to the form $G = [I|-B^T]$ but I just cannot get my solution to match the solution in my notes. I am reducing it Mod 3 as it is part of a coding theory question. So my ...
-1
votes
0answers
12 views

RLC network question (involving matrices / homogeneous systems) [closed]

I need help solving this problem for my homework and I'm not sure how to. What matrix should I be using? I mainly do not know how to set up the problem.
5
votes
2answers
102 views

Prove: $(\det(A-B)+\det(A+B) )^2 \ge 4\det(A^2-B^2 )$

Let $A,B \in \mathcal{M}_n (\mathbb{R})$ two matrices so that: a) $AB^2=B^2 A$ and $BA^2=A^2 B$ b) $\text{rank}(AB-BA)=1$. Prove: $$(\det(A-B)+\det(A+B) )^2≥4\det(A^2-B^2 )$$ ...
0
votes
1answer
26 views

Let $f,g$ be linear operations $V \to V$ such that $\ker f=\ker g$

Let $f,g$ be linear operations $V \to V$ such that $\ker f=\ker g$. Following statement is true: $\text{Im } f=\text{Im }g$ I'm a newbie linear algebra but i know that $\dim \ker f+\dim ...
1
vote
0answers
36 views

Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $BA^2=A^2B$. Prove that $\det(AB-BA)=0$

Problem: Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $BA^2=A^2B$. Prove that $\det(AB-BA)=0$ attempt: $A(AB-BA)=A^2B-ABA=BA^2-ABA=(BA-AB)A=-(AB-BA)A$ So ...
0
votes
0answers
19 views

Find a matrix $P$ for a square matrix $B$ with all entries $(B)_{ij} = b$, $b \in R$. $P$ is a matrix that orthogonally diagonalize matrix $B$.

The condition must meet $$D = (P^{T})BP $$ or $$D=(P^{-1})BP$$ I'm having trouble finding a pattern for all entries and infinite square size matrix. I found a matrix P that is 5x5 with all ...
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1answer
14 views

Trace of a product of two PSD symmetric matrices being zero means this product being a zero matrix?

Some one can help me with this problem? I have two real positive-semidefinite matrices $P$ and $Z$, $P \succeq 0$, $Z \succeq 0$, and they are both symmetric ($P^T = P$ and $Z^T = Z$). Also trace$ ...
2
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0answers
40 views

Are there Latin squares with no repeats on the diagonal not of the form 2y+x+1(mod n)?

I am looking for a certain kind of latin square (nxn). Rules: No repeats in any column or row (Definition of Latin Square) No repeats in any diagonal including others than the main diagonal. So ...
0
votes
0answers
18 views

Reducing generator matrices for the codes $N_5(3,5)$ and $N_5(3,5)^\perp$ to a standard form $[IB]$ or $[BI]$

I am just covering a chapter on MDS codes and am attempting a question but not getting the same solution as in the notes. Here is the question: Write out the generator matrices for the codes ...
0
votes
0answers
19 views

Integral weighted graphs

I found some useful guidelines to investigate integral graphs (i.e. that the eigenvalues of the adjacency matrix are all integers) http://link.springer.com/chapter/10.1007%2FBFb0066434 . However, ...
0
votes
2answers
12 views

Uniqueness of probability given marginals

Let $X,Y,Z$ be finite sets, and consider probability distributions $p$ over $X\times Y\times Z$. If we know the marginals of $p$ over all the pairs $X\times Y$, $X\times Z$ and $Y\times Z$, is that ...
0
votes
1answer
12 views

bounded input, bounded state stability in reverse

I have a stable, linear, time-invariant system with state space representation (A, B, C, $0$). I have established that the state remains bounded, i.e. $x \in \mathcal{L}_{\infty}$. Ironically I have ...
0
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0answers
22 views

Norm bound for the Jordan basis matrix

Suppose I have a matrix $A$. We can assume, if it will be relevant, that it is stochastic with entries in $\{0,\frac{1}{2},\frac{3}{4},1\}$ and eigenvalues with magnitudes in $(0,1]$. $A$ has a ...
0
votes
1answer
20 views

Determine the entries of the matrix given vectors (1,1,1), (1,0,-1), and (1,-1,0) are eigenvectors of the following matrix

M =$\left[\begin{array}{ccc}1& 1 & 1 \\a & b & c \\d & e & f\end{array}\right]$. Work so far: Let $\lambda$ denote the eigenvalues. By my calculations, the eigenvalues must ...
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0answers
17 views

Elementary book on Matrices

can anyone recommend a book that focusses on matrix theory at an elementary level? I was never taught matrices in high school (25 years ago) and I'm teaching myself algebra using The Everything Guide ...
0
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1answer
20 views

Diagonalising an infinite dimensional Hermitian square matrix

I have a quantum state which takes the following form: $\rho = \sum_{b, \,c \, = \,0}^\infty \frac{(-igt)^b(igt)^c}{\sqrt{b!c!}}\vert b\rangle\langle c\vert$. This is an infinite Hermitian matrix ...
0
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1answer
17 views

Trying to prove a rank equation and more

Suppose $A\in\mathbb R^{n\times n}$, $a,b\in\mathbb R$, $a\ne b$ and $$(A+aI_n)(A+bI_n)=0$$ where $I_n$ denotes the identity. Prove 1). $\text{rank}(A+aI_n)+\text{rank}(A+bI_n)=n$ 2). $A$ ...
0
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1answer
49 views

Relation between two inverses

Suppose you know $(I+T)^{-1}$, is there any way for approximate the inverse of the matrix $(I+\alpha T)^{-1}$, where $\alpha\in{\mathbb{R}}$?
3
votes
1answer
83 views

$A^2=AB+BA$. Prove that $\det(AB-BA)=0$

Let $A,B$ be two $3\times 3$ matrices with complex entries, such that $A^2=AB+BA$. Prove that $\det(AB-BA)=0$ Nice problem, and I want to find a solution. $AB-BA=A^2-2BA=(A-2B)A$ so if $|A|=0$ we ...
0
votes
0answers
26 views

Fast algorithm to invert a large sparse matrix

I am interesting in sparse matrix that defined at here. I am looking for a fast algorithm to invert the matrix (better than Gaussian Elimimation). Could you suggest to me some methods that reduce ...
5
votes
6answers
880 views

Do commuting matrices share the same eigenvectors?

In one of my exams I'm asked to prove the following Suppose $A,B\in \mathbb R^{n\times n}$, and $AB=BA$, then $A,B$ share the same eigenvectors. My attempt is let $\xi$ be an eigenvector ...
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votes
0answers
19 views

The Formula for the Equicalent Resistance of Complex Resistance Network(Circuits) [closed]

I have found the formula for the equicalent resistance of general complex resistance network, and written a printable article. So if you are interested in it, you can download it to read from ...
1
vote
1answer
30 views

Can find the determinant of a matrix A of size $n$ in terms of the traces of $A^m$ [duplicate]

We can find the determinant of a matrix A of size $n$ in terms of the traces of $A^m$, for $m=1,…,n$ ? It's det of a matrix with term are traces, but i saw but i can't remember Expressing the ...
0
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1answer
42 views

What will be the eigen values of new matrix?

$A $ is any real symmetric matrix and $\alpha_1,\alpha_2...,\alpha_n$ are eigenvalues of $A$. We are constructing a new matrix $B$ whose diagonal enteries are twice the diagonal enteries of $A$ and ...
0
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1answer
32 views

Symmetric matrix with zero elements below the anti-diagonal

I came across the matrix of the form $$ \begin{pmatrix}a_{1} & a_{2} & a_{3} & a_{4}\\ a_{2} & a_{5} & a_{6} & 0\\ a_{3} & a_{6} & 0 & 0\\ a_{4} & 0 & 0 ...
1
vote
2answers
22 views

Why cannot the homogeneous coodinates be zero?

Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. Why can't Z be zero?
1
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1answer
34 views

number of roots of unity which satisfy a given polynomial

Let $A$ be a matrix over $\mathbb{R}$ and $p_A(x)$ its characteristic polynomial. Is there an easy way to find out how many of the roots of $p_A(x)$ are roots of unity? Fixing a positive integer $k$, ...
1
vote
1answer
33 views

Convert a general equation system to matrix form

I have an equation, where $x$ is a vector of variables and the rest (vector $a$, vector $b$, matrix $M$ and $c_1, c_2, c_3$) are parameters: $a_i - x_i + c_1\sum_{j=1}^{n} m_{ij} x_j - c_2 ...
1
vote
1answer
42 views

Set of all orthogonal matrices over $\mathbb C$ is compact/not

How to show the fact that the set of all orthogonal matrices over $\mathbb C$ is compact By an orthogonal matrix over $\mathbb C$ I mean a matrix $A$ satisfying $AA^T=I$ and here $A^T=(a_{ji})$ where ...
2
votes
1answer
51 views

Exponential of a matrix always converges

I am trying to show that the exponential of a matrix converges for any given square matrix of size $n\times n$: $M\mapsto e^M$ e.g. $\displaystyle e^M = \sum_{n=0}^\infty \frac{M^n}{n!}$ Can I argue ...
0
votes
1answer
35 views

What is this matrixs question asking?

Homework question: There are multiple things I do not understand about this question. What is meant by linear combinations in this context? What is $rb1$ and $sb2$? (what do the r and s represent) ...
1
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1answer
25 views

Subgroups of $GL(2, \Bbb{R})$

I am wondering if a subgroup of $GL(2,\Bbb{R})$ which is constructed by all rotations and all matrices in the form of $$ \left[ \begin{array}{l l} a & x \\ 0 & \sqrt{a} \end{array} \right] \ ...
0
votes
2answers
34 views

$\partial (Q A) = Q (\partial A$) for an orthogonal matrix $Q$?

Let $A \subset \mathbb{R}^d$ and let $Q \in \mathbb{R}^{d \times d}$ be an orthogonal matrix. For a set $B \subset \mathbb{R}^d$, denote $Q B:= \{ Qx : x \in B\}$. Does it hold for the boundary of the ...
1
vote
1answer
41 views

Prove that a ring does not have a multiplicative identity

Let $R =$ the set of all matrices $ \left[ \begin{array}{cc} x & 0 \\ y & 0 \\ \end{array} \right] $ where $x, y \in \mathbb Z$ with R being a ring under matrix addition and ...
1
vote
1answer
26 views

partial differentiation and quotient rule

I want to compute a partial differentiation $\frac{\partial A}{\partial q}$ where A is ($\ddot{q}$), the output of standard manipulator equation, i.e. $$ H(q)\ddot q + C(q,\dot q)\dot q+G(q) =0 $$ ...
0
votes
0answers
9 views

Dimensions of matrices around a diagonal matrix?

The matrices $\mathbf{L}$, $\beta$ and $\mathbf{c}$ are ($j \times b$), ($b \times 1$) and ($j \times 1$) dimensional, respectively, with $j \le b$. The matrix $\mathbf{X}' \mathbf{X}$ is a diagonal ...