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), ...

learn more… | top users | synonyms (2)

1
vote
2answers
20 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 ...
0
votes
0answers
6 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 ...
-1
votes
0answers
30 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) = ...
2
votes
2answers
35 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
2answers
23 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
votes
1answer
7 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 ...
1
vote
1answer
9 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 ...
1
vote
1answer
9 views

Hilbert-Schmidt matrix and square sumable

I am reading a note but I encountered two concepts that I don't know their definitions. 1- Hilbert-Schmidt matrix I searched but I only saw measures and such things, I want definition of ...
0
votes
0answers
4 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
votes
0answers
22 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
vote
0answers
20 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
22 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& ...
0
votes
1answer
14 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. ...
1
vote
1answer
20 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 ...
12
votes
2answers
390 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 ...
0
votes
0answers
15 views

Derivation of energy function

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
-1
votes
0answers
8 views

Frobenius vs Subordinate Matrix norm

I can characterize the columns of a matrix using its norm as $\frac{\Vert A_i\Vert_2}{\Vert A \Vert_F} $, where $A_i$ is a column of $A$. Instead one can also use subordinate matrix norm for this task ...
3
votes
1answer
36 views

How to change of basis from 3 points

I'm a computer scientist and I'm not very good with mathematical stuff... I have got 3 points A, B, C that doesn't create an orthogonal space. I have the coordinates of those 3 points in two 3D ...
1
vote
1answer
47 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 $ ...
2
votes
0answers
31 views

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row?

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row(uniform distribution)? What sort of algorithm should I use to do this task? Brute Force algorithm- ...
0
votes
0answers
26 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
0
votes
1answer
14 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, ...
0
votes
1answer
28 views

Transition Probability matrix for on/off swtiches

Lets say we have $5$ on/off switches. On state is designated by 1 and off state is designated by $0$. Then, $\{ W_k^i, k =0,1,2,\ldots\}$ is a discrete parameter Markov chains (DPMCs) for each ...
0
votes
5answers
28 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 ...
0
votes
0answers
17 views

Compact representation of the following matrix

I have a matrix that has the following structure \begin{bmatrix} a_1(1) & 0 & a_2(1) & 0 \\ 0 & a_2(1) & 0 & a_1(1) \\ a_1(2) & 0 & a_2(2) & 0 \\ 0 & a_2(2) ...
3
votes
1answer
51 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
2
votes
2answers
32 views

Does the lowest diagonal element of a real symmetric matrix form an upper bound to the lowest eigenvalue?

If I have a real symmetric matrix, is it possible to look at the lowest diagonal element and then claim that the lowest eigenvalue of the matrix must be less than or equal to that diagonal element? I ...
0
votes
4answers
97 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 ...
0
votes
2answers
32 views

How to determine if matrices are similar?

Trying to teach myself some Linear Algebra, now trying to study about similar matrices concept, but i am having some trouble (maybe because i am trying to teach myself), found a question online and i ...
0
votes
0answers
15 views

Algorithm for getting Markov chain given the complex eigenvalues

Given real and complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a Markov Chain which has these eigenvalues. I know the Markov chain is not unique as eigenvectors are ...
0
votes
0answers
32 views

a matrix metric

Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ matrices of the same size. We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and $V=\prod_i ...
0
votes
2answers
27 views

Algorithm for real matrix given the complex eigenvalues

Given complex eigenvalues (occurring in conjugate pairs) how to get a single instance of a real matrix which has these eigenvalues. I know the matrix is not unique as eigenvectors are not fixed but in ...
5
votes
2answers
101 views

Eigenvalues of linear operator $F(A) = AB + BA$

Let $B$ be the $n \times n$ square matrix; $\lambda_1, \lambda_2, \dots, \lambda_n$ are its pairwise distinct eigenvalues. For all $n \times n$ matrix $A$ let me define $F(A) = AB + BA$. We can ...
1
vote
1answer
15 views

Constructing matrices with eigenvalues equal to roots of a given polynomial

Suppose we are given a polynomial e.g. $$x^4+Ax^3+Bx^2+Cx+D,\tag1$$ and we need to construct a matrix, whose eigenvalues would be equal to roots of this polynomial. One way, rather inelegant, is to ...
0
votes
1answer
54 views

prove that there exists an upper triangular matrix U such that (U^T)U=A

Let A be a positive definite matrix \begin{pmatrix} a & b \\ b & c \\ \end{pmatrix} prove that there exists an upper triangular matrix U such that U transpose times U equals A. I'm ...
2
votes
1answer
29 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 ...
0
votes
3answers
26 views

$A$ is a doubly stochastic matrix, how about $A^TA$

I am reading a paper with assumption that $A \in R^{n\times n}$ is a doubly stochastic matrix. However, the paper says $A^TA$ is symmetric and stochastic. Since $A^TA$ is symmetric, if $A^TA$ is ...
3
votes
1answer
56 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 ...
1
vote
2answers
26 views

Orthonormal basis for the null space of almost-Householder matrix

A matrix $H$ is defined as: $$H = I - vv^T$$ where $v$ is a unit vector. What is the rank of $H$? What would be an orthonormal basis for the null space of $H$? How do we find the number of zero ...
1
vote
3answers
33 views

Eigenvalues of Householder matrix

What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it to me intuitively or with a simple proof?
0
votes
1answer
23 views

Determine the dimension of $U+W$ and of $U \cap W$. Which sums are direct sums?

Problem: Determine the dimension of the sum $U + W$ and of the intersection $U \cap W$ of the following subspaces $U$ and $W$. Which sums are direct sums? 1) $U = \text{span}\left\{(1,1,1)\right\}$ ...
0
votes
1answer
30 views

Why the Householder matrix is orthogonal?

A Householder matrix $H = I - c u u^T$, where $c$ is a constant and $u$ is a unit vector, always comes out orthogonal and full rank. Why $H$ is orthogonal (looking for an intuitive proof rather than ...
1
vote
0answers
14 views

Hesse matrix is negatively semidefinite if a function has a local maximum

Let $U \subset \mathbb{R}^n$ be open, and let $f: U \to \mathbb{R}$ be at least twice continuously differentiable. Also, we assume $f$ has a local maximum $a \in U$. I now want to show that the Hesse ...
0
votes
0answers
24 views

How to make a $4 \times 6$ matrix from a $6 \times 6$ diagonal matrix in MATLAB [on hold]

I am having a diagonal matrix A = diag(a,b,c,d,e,f) which is a $6 \times 6$ matrix. From this I want to make $4 \times 6$ matrix specified by: ...
0
votes
0answers
48 views

First derivative of energy function [on hold]

Given the following energy function $E(d)$ (also found here on page 3): $$ E_d = \sum_{x,y \in \Omega} \left(d_{x,y} - \hat{d}_{x,y}\right)^2 + \lambda \sum_{x,y} \left( ...
1
vote
2answers
23 views

Is there a way to turn this summation into a matrix multiplication?

I have two vectors $\mathbf{s}, \mathbf{p}$ of length $n$, and I need to compute a vector $\mathbf{\pi}$ defined by $$\pi_i=\sum_{j=1}^is_j(p_j-p_i)$$ for $i$ from $1$ to $n$. I suspect this ...
4
votes
3answers
74 views

Is this operator $A = \pmatrix{1&1\\0&1}$ self-adjoint?

Is this operator $$A = \pmatrix{1&1\\0&1}$$ self-adjoint? I think not, because $$\pmatrix{1&1\\0&1}^T\neq A$$ where $T$ is the transposition of the matrix. What do you all think?
2
votes
1answer
19 views

Number of components needed for 3D rotation

Using Euler angles, a 3D rotation can be expressed using 3 real numbers. Using quaternions, 4 are needed and using rotation matrices 9. Is it possible to express a 3D rotation using less than 3 real ...
5
votes
1answer
59 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 ...
0
votes
1answer
17 views

How to prove infinite solution vs no solution for singular matrix problem.

In the problem Ax=B My coefficient matrix is \begin{bmatrix} α-1 & 1-α\\ α & -α \end{bmatrix} x is \begin{bmatrix} ln K\\ ln L \end{bmatrix} b ...