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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0
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0answers
59 views

Needs an explanation on why I obtain this covariance matrix

Let's say $n$ is an even integer. I'm playing with a column vector $\mathbf{v}$ which must satisfy the following three requirements: It's a length-$n$ vector of +1s and -1s. It has the same number ...
0
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0answers
25 views

quaternions are less versatile than matrix?

I am doing some research looking should I implement quaternions or matrices. What I've seem to come across is that while quaternions can be better for doing smooth rotations and dual quaternions can ...
0
votes
1answer
24 views

Matrix,Linear algebra,polynomial,finite field,notation

In the book by Arora and Barak,Computational Complexity,on page 168,1st paragraph, there is a notation which I do not understand. They write For every $n \times n$ matrix $A$,and $i\in [n]$,we define ...
1
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1answer
42 views

Finding the Jordan Form and basis

$$A= \begin{pmatrix} 2&1&2\\ -1&0&2 \\ 0&0&1 \end{pmatrix}$$ I found that $$f_A(x)=m_A(x) = (x-1)^3.$$ So the Jordan form must be: $$J= \begin{pmatrix} 1&0&0\\ ...
0
votes
3answers
84 views

Prove or disprove: $A=A^\top \land B = B^\top \Rightarrow AB = (AB)^\top$

where $A,B\in\mathbb{R}^{n\times n}$. My current solution is that this will only work iff $A$ and $B$ commute. Since: $(AB)^\top = B^\top A^\top = B A$ $\ $ ($=AB$. iff $A$ and $B$ commute.) I ...
0
votes
1answer
35 views

Are there necessary and sufficient conditions on $A$ and $B$ such that each row of $AB$ has a nonzero entry?

Let $A$ be an $n_A \times n$ matrix and $B$ an $n \times n_B$ matrix. What are necessary and sufficient conditions on $A$ and $B$ such that each row of $AB$ has a nonzero entry?
0
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0answers
28 views

Article writing: How to represent a matrix by its elements?

Intuitively I guess that parantheses with subscript and superscripts is a way of representing a matrix or an array by its elements, e.g., $$ A = (a_{i,j})_{i,j=1}^n $$ (This is taken from here) ...
2
votes
2answers
46 views

Realizing the oscillator algebra as a matrix Lie algebra

In Hilgert's & Neeb's Structure and Geometry of Lie Groups, they introduce a Lie algebra, which they call the "oscillator algebra," as an extension of the Heisenberg algebra. They give a basis ...
0
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0answers
25 views

Increasing Matrix

Consider real matrix-vector multiplication. I am just curious whether there exists a $p\times p$ matrix $A$ such that if $x$ is a $p\times 1$ real vector whose entry is in ascending order, i.e., $x_1 ...
1
vote
4answers
59 views

Find all $2 \times 2$ matrices $A$ and $B$ such that $AB = BA$

Find all possible $2 \times 2$ matrices A that for any $2 \times 2$ matrix B, AB = BA. Hint: AB = BA must hold for all B. Try matrices B that have lots of zero entries. I'm clueless as to how to ...
1
vote
3answers
57 views

If $\mathbf{A}$ is a $2\times 2$ matrix that satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, then $\mathbf{A}$ is invertible

$\mathbf{A}$ is a $2\times 2$ matrix which satisfies $\mathbf{A}^2 - 4\mathbf{A} - 7\mathbf{I} = \mathbf{0}$, where $\mathbf{I}$ is the $2\times 2$ identity matrix. Prove that $\mathbf{A}$ is ...
0
votes
2answers
19 views

Determine if a basis consists of eigenvectors

So, this might be a silly question, but here it is. I am doing a couple of problems computing $[T]_\beta$, and determining whether $\beta$ is a basis consisting of eigenvectors of $T$. My problem is ...
-4
votes
1answer
38 views

prove Determinants are equal [closed]

given matrix $A$, matrix $B$ is the matrix we get after adding row $i$ plus a linear combination of other rows to the row $i$, in matrix $A$. Prove that : $detA=detB$
1
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0answers
18 views

Finding points inside of a box

I have a set of points in 3D that define a large, complex object. These points are rendered in OpenGL for an Android app that I am programming. In this app, the user translates the center of the box ...
0
votes
0answers
18 views

When to use which condition number? (which norm)?

The condition number is used to determine how sensitive b is to changes in A in the equation ...
-1
votes
1answer
52 views

Calculating determinant of matrix $n\times n$ [closed]

matrix M $n\times n$ given, which its left-right diagonal contains the numbers from $1$ to $n$, and all the other numbers equal $n$. calculate the determinant of the matrix M.
6
votes
0answers
63 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
0
votes
1answer
35 views

Linear Recurrence Using matrix exponentiation

Matrix Exponentiation can be used to solve Linear Recurrence . I know how to solve linear recurrences like : $f(n) = f(n-k_1) + f(n-k_2) + ... +\ constant$ But i couldn't find any information on how ...
1
vote
3answers
102 views

Eigenvectors of real symmetric matrices are orthogonal (more discussion)

This is an old question, and the proof is here The proof assumed different eigenvalues with different eigenvectors. My question is how about the repeated root? How to guarantee there will not have ...
3
votes
1answer
18 views

Get amount of submatrixes from $a \times b $matrix

I was trying to do the following exercise Given a grid of size $a \times b$, write a formula able t calculate the total number of rectangles contained in this rectangle. All integer sizes and ...
0
votes
0answers
32 views

What is the minimizer of the matrix norm and it's significance?

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? ...
0
votes
1answer
61 views

eigenvalues and $A^k$

Consider $$A=\begin{pmatrix} s+1 & 1-t \\ -1-t & s-1 \end{pmatrix}$$ where $s$ and $t$ are real numbers. (a) For which values of $s,t$, $A$ is not diagonalizable. (b) For which choice of ...
0
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0answers
22 views

Matrix Multiplication By Rows And Columns

A given matrices $A$ and $B$ produces $A\cdot B=C$. We can say that: $Col(C)=$ linear combinations of the $Col(A)$ $Rows(C)=$ linear combinations of the $Rows(B)$ And $A\cdot[B^1 ...
2
votes
1answer
29 views

Convergence of a sequence of $2\times 2$ real matrices

My Try: So $a_n$ can be written as a series very similar to the taylor series of sin: $\displaystyle a_n=\sum_{k=0}^n \frac{(-1)^k b_k}{(2k+1)!}$ for some $b_k$ to be determined. But it is very ...
0
votes
1answer
13 views

Proof of some properties about orthogonal matrices

$Q$ is an orthogonal matrix, how to prove. $$\langle Qu,Qv\times Qw\rangle=\langle u,v\times w\rangle$$ for any $u,v,w$ which belong to $\mathbb R^3$ Much obliged if you can help me!
3
votes
3answers
41 views

Proving whether linearly independent [closed]

I've been working on this for almost half hour, can someone answer this question perhaps? Thanks. Let {$\vec u_1, \vec u_2, \ldots , \vec u_k$} be a linearly independent set of vectors in ...
0
votes
1answer
24 views

Common solutions to quadratic equations associated to self-adjoint matrices

Let $\mathcal{H}$ be a complex Hilbert space of dimension $d<+\infty$, and let $\{|n\rangle\}$ with $n=0,\cdots,d$ be an orthonormal basis in $\mathcal{H}$. Let $\mathbf{A}$ be a self-adjoint ...
0
votes
0answers
18 views

Minimize residual Matrix

I have two Matrices $A$ and $B$ of order $m\times n$ and Matrix $C=A-B$ I want to formulate a optimisation problem such that I get the difference at each element should go be minimum, ie, a null ...
0
votes
1answer
24 views

Notation for matrix that is partially unknown.

I have a matrix with some elements known and some unknown. I am using the notation $A(X)$ where $X$ are the unknown elements (not sure if relevant but I will be solving for the unknown part $X$ ...
6
votes
1answer
94 views

For $T\in \mathcal L(V)$, we have $\text{adj}(T)T=(\det T)I$.

Let $V$ be an $n$-dimensional vector space over a field of characteristic $0$. For a linear operator $T\in \mathcal L(V)$, we know that $\bigwedge^n T=(\det T)I$, where $I:V\to V$ is the identity map. ...
0
votes
1answer
12 views

Calculate probability from 1 node to another in a adjacency matrix of probabilities

Given a adjacency matrix A like the following: 0.50 0.25 0.25 0.25 0.50 0.25 0.25 0.25 0.50 Where each line i and column j represents a Node and ...
1
vote
1answer
13 views

Prove that an underdetermined system of cannot have a unique solution(Is this proof correct?)

I know I misspelled underdetermine but is this proof correct? How can I improve it either way? Side Remark: Anyone who is down-voting please can you understand I new to this site and somewhat ...
0
votes
0answers
135 views

Solving a BTTB system by BCCB extension that is highly structured and fewer degree of freedom

Consider a BTTB system generated by a simple $3\times 3$ matrix, $$ Col_1 = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ ...
0
votes
0answers
10 views

Singular value decomposition - unique determination

i am self-studying SVD - and stumbled upon the Wikipedia page (https://en.wikipedia.org/wiki/Singular_value_decomposition) on the statement that a common convention is to order the singular values in ...
2
votes
0answers
34 views

Rotating a 3d-ellipse equation?

So I have very limited Linear Algebra knowledge, and I'm trying to program a computer graphics application in Android using OpenGL. I understand my design is not great, so if you have questions as to ...
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votes
0answers
49 views

Is there a compact way to express $(\mathbf{a}^{H}\mathbf{b} + \mathbf{b}^{H}\mathbf{a})$, where $\mathbf{a}$ and $\mathbf{b}$ are complex vectors?

This formula gives twice the inner product of $\mathbf{a}$ and $\mathbf{b}$ if we expand them as real vectors. $H$ is the conjugate transpose.
0
votes
1answer
24 views

how to solve a matrix equation like that

$$M^{T}MY + \lambda Y = D$$ M, Y, D are matrix, $\lambda$ is scalar value, Y and D have the same dimension. M and D is known, how to solve Y.
0
votes
0answers
21 views

Is there any relationship between a worst matrix and its size and what are their common structures?

I am currently trying to test and calculate the worst possible $\mathcal{O}(f(n))$ for some algorithm. In order to do so, I need to find the worst possible (0,1) n x n matrix for some $n$s (e.g. ...
1
vote
2answers
41 views

How to know if it is possible to rearrange columns of a matrix to avoid nulls on the diagonal?

I have a square matrix with binary entries, 1 - 0. Is there a mathematical way/trick/algorithm to know if it is possible to rearrange the columns in order to have all the elements on the main ...
1
vote
0answers
25 views

Analytic Solutions To Matrix Differential Equation

Given the matrix differential equation: $\frac{d U(t)}{dt} = A(t) U(t)$ and the fact that $A_t$ is comprised only of analytic functions Is it possible to conclude that the solution $U(t)$ will be ...
3
votes
4answers
109 views

Find the matrix $\mathbf{A}$ if $A\binom{7}{-1} = \binom{6}{2}.$

Find the $2\times2$ matrix $A$ where $A^2=A$ and $$A\begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$$ I tried plugging in: $A= ...
2
votes
1answer
37 views

Minimizing variance subject to linear inequality

Let A be a $n \times n$ matrix. Where $A$ is a symmetric positive definite matrix. Let $b$ be a vector in $R^n$. $x$ is an unknown vector to be determined. I'm interested to find vector $x$ such ...
0
votes
2answers
15 views

Generalized inverse of a matrix.

Let $A$ be an $m \times n$ matrix of reals with $m\leq n$. I am interested in a generalized inverse $A^+$ of size $m \times m$ such that $A^+A=(I_m \ \bf 0)$. If Rank$(A)=m$, does such an $A^+$ always ...
1
vote
0answers
17 views

Is there a map that maps the mean vector and the variance matrix of a multivariate lognormal to its location vector and diffusion matrix?

Suppose ${\xi} \sim logNormal_d ({\mu},{\Sigma})$, where $\mu$ is a d-dimensional vector (called location vector) and $\Sigma$ is a $d \times d$ symmetric positive definite matrix (called diffusion ...
0
votes
2answers
39 views

Is it possible to get the original eigenvector after scaling a matrix?

Let ${\mathbf{X}}\in\mathbb{R}^{n\times 1}$ and ${\mathbf{Y}}\in\mathbb{R}^{n\times 1}$ and let $\mathbf{A}\in\mathbb{R}^{n\times 2}$ be defined as \begin{equation} \mathbf{A} = ...
0
votes
1answer
31 views

Bounded Matrix-Vector Multiplication

Given a $p\times p$ square matrix $A$. Can I say that the 2 norm of their product is always bounded for any $p \times 1$ vector, please? That is, $$ \| Ax \| <\infty, \forall x\in\mathbb R^p. $$ ...
3
votes
0answers
90 views

Spectral radius of a real, symmetric, positive semi - definite matrix.

While answering a question, the OP made a follow - up question, that I was not able to answer at that moment. However, I came up with an intriguing (at least to me) question. Let ...
0
votes
1answer
32 views

How can I relate two systems of ODEs, when their initial conditions are related?

Consider two simple ODEs with identical right-hand sides, whose initial conditions are related by a simple formula. $$x' = a \cdot x, \quad y' = a \cdot y, \quad x(0) = \alpha \cdot y (0)$$ The ...
1
vote
1answer
44 views

Trace of a real, symmetric positive semi-definite matrix

I have a naive question about the trace of a real, symmetric positive semi-definite matrix: Does the trace of a real, symmetric positive semi-definite matrix have to be larger than $1$? I know ...
1
vote
1answer
29 views

How do the rows of a change of basis matrix form a basis for expressing columns?

I am reading this article on Principal Component Analysis (PCA) and in section III-B (page 3) it has strange definition I don't understand. In the toy example $\mathbf{X}$ is an $m \times n$ ...