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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1answer
19 views

Finding a basis for unification of two subsets

I have this problem : $U,W \subseteq R^4$ Base of $W = \{w1 = (1,2,2,-2), w2 = (0,1,2,-1)\}$ Base of $U = \{u1 = (1,1,0,-1),u2 = (0,1,3,1)\}$ Find a basis for $U \cap W$. My solution for any $v ...
0
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4answers
71 views

Sin(x) + Sin(y)

When you add sound waves you are basically adding sine and cosine of certain multiples of x. Is Sin(x) + Sin(y) ... + Sin(n) = Sin(x+y...+n)? Is the same true for summation of cosines? I am making a ...
1
vote
2answers
56 views

Scalar value of similarity between Two Square Matrices

I am wondering if there is any way to compute mathematically the similarity/distance between two square matrices as a single value?
0
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1answer
15 views

Support of vector $w$ in graph sparsity

I'm reading about graph sparsity and I have one problem in a paper I'm reading I don't understand, maybe someone can clarify: Graph Sparsity: In graph sparsity, we have a directed acyclic graph ...
2
votes
2answers
167 views

Solve the system of linear equations by Gaussian elimination and back-substitution.

Question 1:Form the adjunct matrix and reduce it to echelon form. I dont know how to write matrices here, so i snap a picture of my operation. Did I do it right? Question 2: Use back-substitution to ...
1
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2answers
268 views

How to find transformation matrix which converts matrix to simple standard form

I have a matrix A$$ \left( \begin{array}{ccc} 0 & 1 \\ a^2 & 0\\ \end{array} \right) $$ Using eigen values, I convert it into simple standard form B: $$\left( \begin{array}{ccc} a & 0 \\ ...
1
vote
2answers
69 views

If $AB+BA=0$, then $A^2B^3=B^2A^3$?

If I have a matrix $A$ and $B$ such that $AB+BA=0$ is it true that $A^2B^3=B^2A^3$? I think that it is false.
2
votes
0answers
122 views

Computing a “cheap” upper bound on the norm of the solution to a linear system

Consider the linear system $A x = b$, where $A$ is an invertible, $n \times n$, real matrix. I would like to compute a "cheap" upper bound on the (p-)norm of the solution; i.e. $\|x\|_p$. One can, of ...
4
votes
3answers
79 views

Does a symmetric matrix $A^2$ imply a symmetric $A$?

Does a symmetric matrix $A^2$ imply a symmetric $A$? Any help would be much appreciated.
1
vote
1answer
29 views

Eigenvalue and eigenvector of $A'A$

Suppose that $\mathbf{A}\in\mathrm{R}^{m\times m}$ is a square but not necessarily symmetric matrix whose eigenvalues and eigenvectors are $\lambda_i$ and $\mathbf{x}_i,$ $i = 1,2,\cdots,m$. Is ...
0
votes
1answer
667 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 ...
2
votes
2answers
146 views

Invertible: A non-square matrix?

So I am doing a question were I have the set column matrix 1 = (3, -8, 1) and column matrix 2 = (6, 2, 5) and the question is asking if this is either a bases for R2 or R3. Can I just say that since ...
1
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0answers
42 views

Variance-covariance matrix of a linear regression model

In finding the covariance matrix of a linear regression model I don't understand this step: $$ E[(b-\beta)(b-\beta)']=E[(X'X)^{-1}X'\epsilon\epsilon'X(X'X)^{-1}] $$ where we've been given that $$ ...
3
votes
2answers
441 views

For what values of k and h does this system of equations have a unique solution?

Here's my system of equations: $x−3y+2z=5$ $2x−5y−3z=9$ $−x−y+kz=h$ So I have $ \begin{bmatrix} 1 & -3 & 2 & 5 \\\\ 2 & -5 & -3 & 9 \\\\ -1 & -1 & k& h ...
1
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1answer
45 views

Do all symplectic transformations give rise to skew symmetric matrices?

Suppose that $ \Delta(x,y) = x^T\Delta y $ where $ \Delta$ is a symplectic matrix of form given in https://en.wikipedia.org/wiki/Symplectic_matrix If I define an inner product $ \alpha(x,y) = ...
0
votes
1answer
210 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?
4
votes
1answer
187 views

Proof of the conjecture that the kernel is of dimension 2

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix ...
-1
votes
2answers
49 views

Linear Alegbra - inverse matrix multiplication

I have a general question. If there is a matrix which is inverse and I multiply it by other matrixs which are inverse. Will the result already be reverse matrix? My intonation says is correct, but ...
0
votes
2answers
56 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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0answers
13 views

Monotone operator without symmetry?

A function $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ is monotone with respect to $P = P^\top\succcurlyeq 0$ if $$ \left( f(x) - f(y) \right)^\top P (x-y) \geq 0 $$ for all $x,y$. Now suppose that ...
2
votes
1answer
141 views

“Degrees of freedom” of some low-rank skew-symmetric matrices

Let $n$ be an even integers. Let $r\in \mathbb R^n$ and $e=[1,1,\dots,1]^T$. If $$A = re^T - er^T,$$ then $A\in \mathbb{R}^{n\times n}$ is of rank 2 and skew-symmetric, i.e., $$A = -A^T.$$ This ...
2
votes
1answer
69 views

Parallelism in Golub & van Loan's Jacobi algorithm for symmetric eigenvalue problems

In Matrix Computations by Golub and Van Loan (3rd edition, page 433) an algorithm is given for a parallel version of the classical Jacobi algorithm for solving a real symmetric eigenvalue problem. The ...
0
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2answers
21 views

Iteration of a function related to the minimal polynomial of a matrix

Let $M$ be a singular $n \times n$ matrix over some field. In order to find a matrix $N$ s.t. $MN=0$, I do the following : $p(x)=$ minimal polynomial of $M$. Then the constant term of p is zero ...
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1answer
35 views

What does this matrix operation mean?

If T is matrix what is this operation? What's name of operation?
2
votes
1answer
367 views

Jacobian Matrix Requirement for Linear Approximation

It is my understanding that when searching for a linear approximation of a nonlinear function, using the Jacobian (matrix) could help. I did some reading, and read that there is a condition: the ...
2
votes
2answers
49 views

Preservation of rank implies Invertibility

Show that if the rank of $XY$ (where $Y$ is an $n\times n$ matrix) is the same as the rank of $X$ for every $m\times n$ matrix $X$, then $Y$ is invertible. I thought I had found a counterexample: $$ ...
4
votes
1answer
96 views

Prove that the kernel is of dimension 2

"Experimentally", I found that the kernel (null space) of the following matrix is of dimension 2. I'd like to prove it, but haven't managed yet: \begin{equation} \text{for almost all } t>0,\quad ...
1
vote
1answer
107 views

Negative determinant

Let $$ A = \begin{bmatrix} -a_{12}-a_{13}-a_{14} & a_{12} & a_{13} & 1\\ a_{21} & -a_{21}-a_{23}-a_{24} & a_{23} & 1\\ a_{31} & a_{32} & -a_{31} - a_{32} - a_{34} & ...
0
votes
1answer
21 views

Help understanding formula $score(K) = \sum_{i,j} | d_{ij} - e_{ij} |$

I am trying to write some code to perform an equation based on the formula below, however I am having a hard time understanding mathematic syntax. The formula is as follows: $$ score(K) = ...
1
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0answers
517 views

Suppose that A and B are nxn matrices with A row equivalent to B. Prove that if A is nonsingular, then B is row equivalent to In.

Suppose that A and B are nxn matrices with A row equivalent to B. Prove that if A is nonsingular, then B is row equivalent to In. I answered this on a test and it seemed right to me, but got zero ...
1
vote
1answer
285 views

Why does the discrete cosine transform as matrix multiplication work this way?

I have read that the DCT can be computed as a matrix multiplication. The 8x8 DCT matrix is: $D=\frac{1}{2}\left[\matrix{ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & ...
1
vote
3answers
549 views

Determinant of linear transformation

Given a linear transformation $T:V\rightarrow V$ on a finite-dimensional vector space $V$, we define its determinant as $\det([T]_{\mathcal{B}})$, where $[T]_{\mathcal{B}}$ is the (square) matrix ...
3
votes
3answers
89 views

If $A=\pmatrix{1 &0\\-1&1}$, show that $A^2-2A+I_2=0$. Hence find $A^{50}$

If $$A=\pmatrix{1 &0\\-1&1},$$ show that $$A^2-2A+I_2=0,$$ where $I_{2}$ is the $2x2$ Identity matrix. Hence find $A^{50}$. We have $$A^2-2A+I_2=A(A-2I_3)+I_=\pmatrix{1 ...
0
votes
1answer
34 views

how to differentiate this equation (contains absolute and norm)

how can I differentiate the following wrt $\mathbf{d}_i$? $\frac{|\mathbf{d}_i^T\mathbf{d}_j|}{\|\mathbf{d}_i\|_2\|\mathbf{d}_j\|_2}$ Thanks in advance.
0
votes
1answer
33 views

Can a general time-dependent finite-dimensional Schrödinger equation with complex Hamiltonian be transformed to one with real Hamiltonian?

Consider a general-form time-dependent Schrödinger equation: $$i\partial_tv=\hat Hv,$$ where Hamiltonian $\hat H$ is an Hermitian matrix (finite-dimensional for simplicity), and $v(t)$ is a complex ...
3
votes
1answer
71 views

Determinant of the matrix $\binom{m_i}{j-1}$

Let $m_1,\dots,m_n$ be real numbers $\ge n-1$. How can I find the determinant of the matrix $A$ defined by $(a_{i,j})=\binom{m_i}{j-1}$, for $1\le i\le n$ and $1 \le j \le n$ ? This all looks ...
0
votes
2answers
458 views

Checking if a matrix is positive semidefinite?

I am trying to figure out if this 2x2 matrix is positive SD. $x$ is in $R_{++}$ and $y$ is in R $\begin{bmatrix}\frac{2}{x} \frac{-2y}{x^2} \\\frac{-2y}{x^2} \frac{2y^2}{x^3}\end{bmatrix}$ A matrix ...
0
votes
3answers
76 views

Linear Algebra - eigenvalue and eigenvectors

I have two questions which I have trouble to prove/disprove. 1) I have trouble to prove this: $A$ is $n \times n$ matrix, if $A^2=A$ then A has at least one eigenvalue. 2) I have trouble to ...
1
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0answers
43 views

Transpose matrix of a linear transformation

Let $T: V\to W$ be a linear transformation and let {$v_1,...,v_n$}, {$w_1,...,w_m$} be ordered basis of $V$ and $W$ then $$ \left\{ \begin{array}{c} a_{11}w_1+a_{12}w_2+\cdots+a_{1m}w_m=T(v_1) \\ ...
1
vote
1answer
36 views

Reduced Row Echelon Form in $\Bbb Z/3\Bbb Z$?

I'm trying to understand the best way to approach this problem. Short of writing every combination of matrices, I'm wondering if anyone can help me learn how to solve this problem. How many $3\times ...
1
vote
1answer
41 views

Meaning of this matrix structure

A professor is keep using the following kind of matrix. The problem is that I have never seen a matrix separated with these lines inside it. What does that kind of matrix mean (not the contents but ...
10
votes
3answers
175 views

Determinant of $a_{i,j}=(x_i+y_j)^k$

How can I find the determinant of the matrix $A\in\mathcal{M}_n(\mathbb{R})$ with coefficients $a_{i,j}=(x_i+y_j)^k,k<n$ ? All the $x_u,y_u$ are real numbers. Derivating won't help, and I didn't ...
5
votes
1answer
67 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
0
votes
1answer
189 views

Eigenvectors for shear matrix and diagonalizing.

Here is a shear matrix $ \begin{pmatrix} 1 && 0 \\ 2 && 1 \end{pmatrix}$. The eigenvalues are 1. $ \lambda^2 - 2 \lambda + 1 \to \lambda = 1$. So now I try to find the eigenvectors. ...
0
votes
0answers
52 views

Eigenvectors of positive matrix

Let $A$ be a real symmetric matrix with positive coefficients. How can we prove that: There exists a positive eigenvector $v>0$ (all $v_i>0$) associated with the greates-absolute-value ...
1
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2answers
132 views

For what value of $x$ is this matrix invertible?

I've been given the following matrix $X$: $$X= \begin{bmatrix} 1 & 4 & 8 & 1 \\ 0 & 30 &1 & 0 \\ 0 &2& 0& 0 \\ 1 &2 & 9 & x \\ ...
0
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3answers
54 views

Inverse of finite squared matrices.

I've usually used that given a square matrix $A$ with determinant $\det(A)\neq0$, then its inverse $A^{-1}$ is the matrix that meets: $$A^{-1}A=\mathbb{I}$$ and $$AA^{-1}=\mathbb{I}.$$ However, ...
1
vote
0answers
42 views

how to derive the product of singular values of this matrix

Well, I know this matrix is too complex, but still any hints or ideas on working it out will be greatly appreciated. To describe this problem, I have to define some notations beforehand. Twsit: ...
1
vote
1answer
83 views

Matrix over GF(2)

Let B be a square matrix, let I be identity matrix of the same size, and let G be the generator matrix in standard form created by appending B to I. Prove that the code over GF(2) generated by G is ...
0
votes
1answer
53 views

Confusion on Eigenvalues of Matrix

I'm a TA with Advanced Algebra in school and teach the Jordan Form now. There are three questions about eigenvalues in this chapter: Given matrix $A$, $B$ and polynome $f$, consider the eigenvalues' ...