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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0answers
48 views

To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz [duplicate]

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
4
votes
1answer
190 views

Calculating the rank of a Boolean matrix and Boolean matrix factorization

I am interesting in some sort of algorithm for calculating the Boolean rank of small $M \times N$ Boolean matrices. Just to be clear, by Boolean matrices I mean matrices with entries $0$ or $1$ where ...
0
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1answer
140 views

Positive definiteness and eigenvalue inequalities of two symmetric matrices

Let $A = A^T \in \mathbb{R}^{n \times n}$ and $B = B^T \in \mathbb{R}^{n \times n}$ be any symmetric matrices. Then, I know that if $A$ and $B$ are positive definite, and $A-B$ is positive semi-...
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6answers
261 views

Gradient of $x^{T}Ax$

I just came across the following: $\nabla x^TAx = 2Ax$ Which seems like as good of a guess as any, but it certainly wasn't discussed in either my linear algebra class or my multivar calc class. Is ...
0
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2answers
119 views

Deriving a Formula for the determinant of a block matrix.

This is a follow up question to this. I want to solve the following problem: Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space n_1+n_2=n$...
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3answers
101 views

Book for Linear Algebra and Matrix

my major is Electrical Engineering and I am new in linear algebra and I need to be familiar with matrix theory deeply because of my research topic which is Image Processing. But, I do not know from ...
1
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2answers
1k views

Limit of a sequence of matrices

I'm preparing or my exam in linear algebra and I'm stuck with a question. I've tried to find some help in my textbook (Linear Algebra and its applications, 4th Edition, By David C. Lay). I can't find ...
9
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1answer
232 views

Determining sign(det(A)) for nearly-singular matrix A

Motivation: determining whether a point $p$ is above or below a plane $\pi$, which is defined by $d$ points, in a $d$-dimensional space, is equivalent to computing the sign of a determinant of a ...
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1answer
47 views

When is the matrix $\mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T}$ a symmetric matrix? [closed]

let $\mathbf{A}\in\mathbb{R}^{m\times n}$ and $\mathbf{x}\in\mathbb{R}^{n\times 1}$. \begin{equation} \mathbf{Y}=\mathbf{A}\mathbf{x}\mathbf{x}^{T}\mathbf{A}^{T} \end{equation} Can we say that $\...
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1answer
51 views

Matrices inside matrix. Showing $det(M)=det(C)$

Let $n \in \Bbb N \space \text{/{0}} \space \text{and} \space n_1,n_2 \in \Bbb N \space \text{such that} \space n_1+n_2=n$ $$M=\begin{pmatrix}E_{n_1}&B\\O&C\end{pmatrix}$$ where $E_{n_1} \...
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1answer
41 views

Question on normal matrices

Hello all I was given this question in my linear algebra class which I have tried to solve but to no avail, and I would really appreciate any help. I am given a matrix $ A \in M_{nxn}(C) $ and am ...
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4answers
56 views

How to find a real matrix with complex eigenvalues,

Give a $2 \times 2$ real matrix $A$ with eigenvalues $2+3i$, $2-3i$. I would like hints only. So far, I've been trying get somewhere with $\det[A-(2+3i)I] = 0$ and $\det[A-(2-3i)I] = 0$; which ...
1
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3answers
106 views

Eigenvalues of matrix $A^TA+I$ are real and greater than 1?

In this paper, the author states that the eigenvalues of the matrix $A^TA + I$ are real and greater than 1, since $A^TA$ is symmetric positive definite. But why?
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2answers
260 views

Get the camera transformation matrix (Camera pose, not view matrix)

Let's say that I have an object and a camera (its representation) in a 3D world coordinate system. I have the camera pose to see the object (rotation matrix and translation (eye position)). If I apply ...
0
votes
1answer
48 views

Bounding the smallest eigenvalue of symmetric matrix product

Let $X = ABA^T$ where $B \in \mathbb{R}^{p \times p}$ and $B$ is positive definite matrix and $A \in \mathbb{R}^{q \times p}$ so that $X \in \mathbb{R}^{q \times q}$. My question is concerning an ...
1
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1answer
72 views

is this an orthogonal matrix?

$T$ is a $4\times 4$ real matrix, and obeys $$T^\dagger \left( \begin{array}{cccc} a & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & a^2 & 0 \\ 0 & 0 & 0 &...
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0answers
20 views

What formula would I use for a four factor prioritization method where the factors are summed and ranked?

We are developing a way to prioritize system issues. Our current ranking is 1 - 5, but that becomes rather flat when dealing with a couple hundred issues. In our new method, we have four factors in ...
1
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2answers
233 views

Finding the scalar derivative of a matrix product

I'm trying to find $$\frac{\partial}{\partial \lambda}y^T \left(\sigma^2 I + \lambda^{-1}K_{\theta}^{-1}\right)^{-1}y$$ where $y \in \mathbb{R^n}$ is fixed, $\lambda \in \mathbb{R}$ and $K_{\theta}^{-...
12
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4answers
1k views

Solving a system of non-linear equations

Let $$(\star)\begin{cases} \begin{vmatrix} x&y\\ z&x\\ \end{vmatrix}=1, \\ \begin{vmatrix} y&z\\ x&y\\ \end{vmatrix}=2, \\ \begin{vmatrix} z&x\\ y&z\\ \end{...
1
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0answers
289 views

Quick way to determine congruences

Is there a quick way to determine if a $2\times 2$ matrix, $M\in M_2(\mathbb R)$, is congruent to $I_2$ over $\mathbb R, \mathbb C, \mathbb Q$? Without explicitly finding the matrices $P\in M_2$ s.t. $...
0
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0answers
17 views

Low rank approximation of a set of matrices with equal residuals

Given a set of $n \times n$ matrices $\boldsymbol{H}_i$ ($i=1,2,\cdots,N$, $N \geq 2$) and an interger $k$ ($k<n$), is there a non-iterative way to solve the following problem if the solution ...
1
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1answer
40 views

Does $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({AY+YA}^T) = \{0\} $ imply $ \mathcal{R}({C^T})\, \cap\, \mathcal{N}({CAY}) = \{0\} $?

Given $ \mathbf{Y}=\mathbf{Y}^T \in \mathbb{R}^{n\times n} >0, \mathbf{A} \in \mathbb{R}^{n\times n} $ Hurwitz, $ \mathbf{C} \in \mathbb{R}^{m\times n}, \mathrm{rank}(\mathbf{C})=m,\ m \le n $, I ...
2
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0answers
19 views

Finding minimal projections in subalgebra generated by a given set

Consider the set of complex matrices $\mathbb{C}^{n\times n}$ for some set. Suppose we have a set $\{A_1,\ldots, A_n\}$ of Hermitian matrices. We want to find minimal projections in the subalgebra $\...
0
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0answers
112 views

Specific vectors in the null space of a complex matrix

I'm dealing here with the following problem: I have a complex matrix $F$, having size $N \times M$, with $N<<M$. I can easily compute a basis of the null space of $F$, i.e. the space of vectors $...
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2answers
258 views

Reduced row echelon form without introducing fractions at any intermediate stage

How can I reduce this matrix to reduced row echelon form but without using fractions in intermediary steps (I can use them in elementary row operations just not in the results in the matrix) $$ ...
1
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2answers
43 views

Proving existence and uniqueness of a matrix,

Let A be nxn with real coefficients and assume that it has n distinct eigenvalues, and all eigenvalues are positive real numbers. Let k $\ge$3 be an odd integer. a) Prove there exists a unique real ...
1
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2answers
78 views

Incomplete Circulant matrix

The eigenvectors and eigenvalues of a Circulant matrix are well-known to be related to the discrete Fourier transform of entries of one row (the exact terms are given here). My question: is there any ...
0
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0answers
83 views

Is there a function that defines the trace of a matrix A divided by its determinant?

Given a matrix $A$ that is $n \times n$ with a non-zero determinant does there exist a function $f(n,A)$ in any field of mathematics such that: $$ f(n,A) = \dfrac{\mbox{tr}(...
0
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0answers
21 views

Calculate camera view and projection matricies from projected points

I’m stuck on a project for a client.. I need to find the answer to this to proceed: Given (n) coordinates in 3D space and (n) corresponding coordinates in 2D space as projected onto a camera’s image ...
2
votes
2answers
64 views

What is the correct way to write this matrix equation?

Given an $n \times m$ matrix $X$ and $m \times m$ matrix $A$, I would like to define the vector $y$ as $$y_i = X_{i,*} A (X_{i,*})^T$$ where $X_{i,*}$ is the $i$th row of $X$. Is there a simpler ...
0
votes
1answer
25 views

The Maximum Eigenvalue of $F\mathrm{max(B)}F^T - FBF^T$

$F$ is a $b \times n$ real matrix. $B$ is a $n \times n$ real matrix, constructed by $B = w^T w$, where $w$ is a row vector with strictly positive real numbers, and clearly $B$ is a rank 1 matrix. ...
2
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0answers
48 views

How does one evaluate the derivative of a matrix with a tensor $\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}$?

I am stuck on the following: $$\frac{\partial \operatorname{Tr}[A(\mathrm{Id}\otimes w)]}{\partial w}=\text{ ?}$$ with $A$ a $d\times d^2$ matrix, $\mathrm{Id}$ the identity matrix of $d\times d$ ...
0
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2answers
62 views

Block Matrix Zero Determinant Implication?

Recently I've been working with a number of square (order of 2n) matrices whose determinants are zero. That is, $$\det\begin{bmatrix}A&B\\C &D \end{bmatrix} = 0$$ where each of A,B,C, and D ...
0
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4answers
10k views

How to prove $A+ A^T$ symmetric, $A-A^T$ skew-symmetric.

Prove that if $A$ is a square matrix, then: a) $A+ A^T$ is symmetric. b) $A-A^T$ is skew-symmetric. c) Use part (a) and (b) to show $A$ can be written as the sum of a symmetric matrix $B$ and a ...
3
votes
1answer
35 views

Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$-independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I am ...
0
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1answer
153 views

Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...
13
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2answers
695 views

Are eigenvalues of the limit of a sequence of matrices limits of eigenvalue sequences?

Let $\{A_n\}\in \mathbb{R}^{m\times m}$ be a sequence of symmetric matrices such that $A_n\to A$ as $n\to \infty$, i.e. $\lim_{n\to \infty}a_{ij}(n)=a_{ij}\ \forall 1\le i,j\le m$ where $A_n=[a_{ij}(n)...
1
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1answer
25 views

How does this form of Poincare's inequality for self-adjoint matrices hold?

I'm reading "Introduction to Matrix Analysis and Applications" by Hiai and Petz, and they state Theorem 1.26 ("Poincare's Inequality") as follows: Let $A\in B(H)$ be a self-adjoint operator with ...
0
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1answer
140 views

Practical examples/implementation details for Gauss-Seidel method

I'm having a presentation on Gauss-Seidel iterative method, and although it isn't mandatory , I would like to have some practical examples for this method (a system of linear equations with n>=1000, ...
0
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0answers
38 views

AAK theorem for finite dimensional Hankel matrix

Does the AAK theorem hold for finite dimensional Hankel matrix? Or maybe similar analysis exists? (From a quick look of the proof, it seems like the AAK solution has to be infinite dimensional ...
0
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0answers
72 views

Given a matrix M, is there a name for the matrix MM^T?

One can make a symmetric square matrix out of any m-by-n matrix $M$ by computing the matrix $MM^T$ (or $M^T M$). Is there a name for this operation? I want to call it "symmetrizing" the matrix, but I ...
1
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0answers
63 views

transofrmations (a,b,c) to (x,y,z)

I'm not 100% sure linear algebra will crunch this problem, but hopefully so. This may just be a case of matrices, which would be good cause I like those. Imagine we have a robot with a camera ...
3
votes
1answer
53 views

Find the inverse of a specific Vandermonde matrix

Let $$ V=\begin{bmatrix} 1& 1& 1& \cdots& 1 \\ 1& \xi& \xi^{2}& \cdots& \xi^{n-1} \\ 1& \xi^{...
2
votes
1answer
707 views

Definition of Reducible matrix and relation with not strongly connected digraph

I connot quite understand the definition of reducible matrix here. We know $A_{n\times n}$ is reducible, when there exists a permutation matrix $\textbf{P}$ such that: $$P^TAP=\begin{bmatrix}X &...
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1answer
52 views

Construction of a matrix over $ \{-1,0,1\} $

Let $ Z=(z_{ij}) $ be a $ (n,n)$-matrix, for which: $ z_{ij} \in \mathbb{R}; $ $ z_{ij}= -z_{ji} $ for $ i,j=1, \dots , n; $ $ \sum_{j=1}^n z_{ij} = 0 $ and $ \sum_{j=1}^n |z_{ij} |>0 $ for $ ...
5
votes
1answer
109 views

Homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$ are conjugate

Let $\phi_1$ and $\phi_2$ be two ring homomorphisms from $\mathbb{C}$ to $M_2(\mathbb{R})$. Show that there exists $g\in GL_2(\mathbb{R})$ such that $\phi_2(x) = g\phi_1(x)g^{-1}$ for all $x\in\mathbb{...
7
votes
2answers
127 views

why similarity over $\bar{\mathbb{F}}$ of $A,B\in M_n(\mathbb{F})$ implies similarity over $\mathbb{F}$?

A classic problem in linear algebra is to determine if two matrices $A,B\in M_n(\mathbb{F})$ are similar one to another. When $\mathbb{F}=\bar{\mathbb{F}}$, we know that $A,B$ are similar if and only ...
0
votes
1answer
24 views

For which value of m are these 3 vectors linearly dependent?

In one of my revision worksheets there is a question which goes as follows: The vectors u=mi+j+k, v=i+mj+k and w=i+j+mk, where m is a real constant, are linearly dependent for either m=0, m=1, m=2, m=...
0
votes
4answers
168 views

I have discovered a way to calculate the absolute value (area,volume, etc) of a n-dimentional shape, using it's coordinates only, how do I publish it?

Firstly, I want to preface by saying that I am no experience with the maths community at all, however I did take Maths and Further Maths for my A-Levels. What I have discovered is a way of using ...
1
vote
5answers
39 views

Find a matrix which maximizes expression

Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$ A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy $$ How can I find the matrix $A$ that maximizes expression $K(A)...