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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2answers
65 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
115 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
78 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.
0
votes
1answer
26 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
620 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
124 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
38 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
357 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) = ...
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1answer
178 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
186 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?
0
votes
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
98 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
62 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
votes
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 ...
1
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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
320 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
46 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
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1answer
87 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
20 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
442 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
224 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
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3answers
288 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
33 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
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1answer
32 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
70 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
230 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
41 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
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1answer
35 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
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1answer
40 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
169 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
66 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
141 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
49 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
126 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 \\ ...
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votes
3answers
53 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
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0answers
41 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
77 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
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1answer
50 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' ...
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0answers
35 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
votes
1answer
81 views

Find values of “a” so “A” eigenvalues absolute values are 1 or less. Applicable theorem?

I´m taking linear algebra, and the professor asked us to get back next class with many methods to solve this exercise. I can´t find even one after 2 hours of thinking, really sad. Could you please ...
0
votes
1answer
402 views

Finding the area of an ellipse using linear algebra

I am new to linear algebra. If my answer is incorrect tips would be appreciated! Q1: Find the area of an ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$ with linear algebra. My work: $b = 5$ ...
1
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2answers
55 views

Matrix Algebra calculation

I am new to linear algebra. If my answer is wrong could you give me some tips. Any tips would be appreciated. Thanks! Q: If A^2 − 2A + I = 0, show A^3 = 3A − 2I What I did: A^2 = 2A - I A(A^2) = ...
0
votes
1answer
73 views

Range of a Linear Transformation

Let $$T : P_4 \rightarrow P_{3} $$ be given by : $$ T(a_0 + a_1x + a_2x^2 + a_3x^3) = (a_0-a_1+2a_2-a_3) + (-a_0+3a_1 - 2a_2+3a_3)x + (2a_0 - 3a_1+ 5a_2)x^2 + (3a_0 - a_1 + 7a_2 + 2a_3)x^3 $$ Find a ...
0
votes
2answers
70 views

Write down an inverse matrix in terms of a matrix

$B^3-2B^2+3B=0$ I need to write it down in terms of $B^{-1}$ (inverse of it) I know how that $BB^{-1}=I$ but seems it wont help to find $B^{-1}$ what kind of method do i have to use ?