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
38 views

About matrices and the nilpotent property.

If an $n\times n$ square matrix $W$ has $r$ of its entries of value of zero (where $1 < r < n^2$) does there exist an integer $t > 1$ such that $W^t$ has $s$ entries being zero, with $s > ...
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1answer
52 views

Notion of Matrices over finite field

Suppose $S$ is a set of all possible square matrices over a finite field $F_p$. What will the notion of determinant, rank, nullity, eigenvalue, eigenvector, adjoint, inverse of the matrix of $S$? I ...
1
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0answers
32 views

Matrix fraction decomposition of an integral

I understand this can be solved using matrix fraction decomposition. However I could not find any information on this topic. Please help. $$ \int_r^s e^{-At}BB' (e^{-At})'dt\,. $$ Where $A$ is an $n ...
0
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0answers
48 views

Representing natural numbers as matrices by use of $\otimes$

What I am wanting to do is to find a unique matrix representations for Natural numbers. Say I have the number $n$, how can I represent this number as a matrix in which I can do matrix multiplication ...
1
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1answer
56 views

Linear Algebra Confusion!

is the set of all upper triangular matrices a vector space? I have tried to research if it's a vector space or a subspace but Linear Algebra is starting to look like foreign language for me. Can ...
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1answer
18 views

How to prove Penrose conditions

Can't even figure out how to tackle this problem, just looking for help with the first part.
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1answer
25 views

How is it possible to solve for singular values of a matrix and how is it different than solving for eigen values?

I am in the process of teaching myself about singular values, SVD and eigenvects.. etc. I am looking at a question asking to find the singular values of a $2\times 3$ matrix, but am unsure what this ...
2
votes
1answer
107 views

Polynomial of matrix

The question here is that, is it possible to solve a polynomial of matrix like the following $A^{2}+A=B$, where $B$ is a known semi-definite matrix, and $A$ is the unknown symmetric matrix we ...
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1answer
28 views

A Matrix with Every Second Entry 0

Suppose I have a matrix $M\in\mathbb{R}^{N\times N}$ such that $M_{ij} = 0$ if $i$ and $j$ are of odd parity (that is to say that if $i$ is even, then $j$ is odd and vice versa). As such, my matrix ...
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0answers
90 views

Prove matrix is nilpotent, find its invariants and Jordan form

Prove that the matrix $A = \begin{pmatrix} 1&1&1 \\ -1&-1&-1 \\ 1&1&0 \end{pmatrix}$ is nilpotent, and find its invariants and Jordan form. So far, I've verified directly ...
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1answer
23 views

Matrix representations of linear transformation between bases

Let V and W be vector spaces, and let L: V -> W be a linear transformation between them. A basis for V is E = {$v_1$,...,$v_5$}. A basis for W is F = {$w_1$,...,$w_4$}. On the basis vectors the linear ...
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0answers
22 views

Representing numbers in unique p-adic to matrix representation, is there a way?

What I am wanting to do, if it is possible is find unique matrix representations for the p-adic representation of numbers. So for example, say I have the number 1365 = 3*5*7*13. Now I could take ...
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1answer
29 views

Creating a matrix such that all the sub-matrices are max rank

Let $A\odot B$ denote the elementwise multiplication of matrices $A$ and $B$. Given a binary matrix $B_{m \times n}=[b_{ij}]$, $b_{ij} \in \{0,1\}$, I want to find a matrix $A=[a_{ij}]$, $a_{ij}\in ...
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votes
3answers
402 views

Are all Vectors of a Basis Orthogonal?

Assuming we have a basis for a set $\mathbb{R}^n$, would any set of linearly independent vectors that form a basis for $\mathbb{R}^n$ also be orthogonal to each other? Take the trivial case of ...
0
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1answer
58 views

Prove that the group of $3\times3$ rational unipotent triangular matrices modulo its center is isomorphic to the additive group $\mathbb Q^2$

Let $G$ be the group of matrices of the form: $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} \right)$$ with $a,b,c \in ...
0
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0answers
86 views

Derivative of quadratic form w.r.t. matrix (product)

I need to show that some quadratic from: 1' A C A 1 is increasing in matrix C , where 1 is a (Kx1) vector of ones, and A and C are both (KxK) positive definite. Can I reason like this: 1) ...
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0answers
18 views

Criterion of removal of equations from overdetermined system

Consider the problem of solving overdetermined system Ax = b; In the problem I am trying to solve (from the field of spectral unmixing) number of unknowns usually varies between N = 2 and 5 and the ...
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1answer
67 views

Spectral Decomposition proof

Spectral Decomposition. Prove that if A is symmetric, and orthogonally diagonalized by P = [u1 · · · un], then $A =\sum_{k=1}^n \lambda_k \, u_k \,u_k^T$ where the $\lambda_k$ are the ...
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1answer
33 views

Matrix Similarity proofs

Say that $A$ and $B$ are similar i.e. there exists an invertible matrix $P$ such that $A = P^{−1}BP$. Prove that: (a) $tr(A) = tr(B)$ (b) $|A| = |B|$ (Notation: for an $n × n$ matrix $M$, $|M| = ...
3
votes
1answer
52 views

Linear Algebra quesion

$A^{-1} - \lambda A = B^{-1} - \lambda B - \alpha v v^T$ $A, B \in S^n_+$; $v \in R^n$; $\lambda, \alpha \in R_+$. Can we solve A in term of other variables?
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0answers
45 views

Extreme points by intersection of extreme rays and hyperplane

I just met one question and have no idea about the proof, hope someone can give me some ideas on how to attack this question. Given a graph $G=(V,E)$ with $|E|=n$. Define a set $S$: ...
2
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0answers
51 views

Powers of (large) lower triangular matrix

Consider the following "game" of chance. Each time the player pushes a button he is awarded a random (finite, integer, non-negative) number of points. The probability of receiving any particular score ...
2
votes
1answer
69 views

Elementary matrix inequality

Let $A \in \mathbb{R}^{n \times n}$ be a positive semidefinite matrix. Is the mapping $$ \begin{align} F \ \colon \ \mathbb{R}^{n \times n} &\to \mathbb{R}^{n \times n} \\ X &\mapsto X^{-1} - ...
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1answer
107 views

condition number with respect to spectral norm

I would like to show that "the condition number for inversion of $A$, with respect to the spectral norm is $k(A)=\rho(A)\rho(A^{-1})$" for $A\in M_n$ as nonsingular and normal matrix . Can anyone ...
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1answer
25 views

About reducible matrices

I'm doing a work about Perron-Frobenius theorem, and I'm trying to give a proof of it. I'm stuck, because I found that an irreducible matrix can't have a row or a column of zeros. I understand that ...
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1answer
28 views

suggest me a example for non singular, conjugate-symmetric sesquilinear form ????

I only know that fact that the matrix corresponding to the non singular, conjugate-symmetric sesquilinear form is a unitary matrix. and SU_n(q) is the unitary groups is the collection of the ...
2
votes
1answer
101 views

Derivative of Hadamard product

What is the derivative of Hadamard product of two matrices with respect to one of them? I.e. what is $D(AB)$ with respect to $A$?
2
votes
2answers
112 views

On existence and uniqueness of non-trivial solution of matrix equation $AX=XA$

Now I have a matrix equation $AX-XA=0$, where $A=A^T$ is real symmetric and $X=-X^T$ is unknown and skew-symmetric. I have transformed the equation into the following, $$(I_n\otimes S-S\otimes ...
5
votes
2answers
83 views

Geometry of Commuting Hermitian Matrices

I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of ...
0
votes
1answer
32 views

Is there a general name for matrices which only have zeros on their main diagonal?

A diagonal matrix is one where every component not on the main diagonal is zero. E.g. $$ \begin{array}{cc} 12 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -2 \end{array} $$ Is there a term ...
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1answer
37 views

Is the cone of rank one matrix convex and closed?

Define the following cone: $M=\{xx^T:x\in\mathbb{R}^n\}$ Is this cone convex and closed? How to prove? Thanks
2
votes
1answer
33 views

Prove that determinant of matrix equal to n

Prove that determinant of matrix $D_n$ (square $n$ x $n$ matrix) is equal to $n$. $$ \begin{matrix} 1 & -1 & -1 & \cdots & -1 \\ 1 & 1 & & & \\ 1 & & 1 & ...
1
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3answers
65 views

Is the set of diagonalizable, complex matrices open in the set of square matrices?

Is the set of complex, diagonalizable matrices open in the set of square matrices? I asked myself this question and I tried to prove it somehow. However, I don't have any good approach so far. ...
2
votes
3answers
48 views

Conditions on certain entries of a matrix to ensure one Jordan block per eigenvalue

In preparation for a future exam, I found the following problem: Let $$A = \begin{pmatrix} 1 & 0 & a & b \\ 0 & 1 & 0 & 0 \\ 0 & c & 3 & -2 \\ 0 & d & 2 ...
0
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0answers
19 views

Which cut-off for collapsing this tree?

I have a Newick tree that is built by comparing similarity (euclidean distance) of Position Weight Matrices (PWMs or PSSMs) of DNA regulatory motifs that are ~5-9 bp long sequences. An interactive ...
2
votes
1answer
53 views

Proving that a Matrix is Invertible

Let $A$ be a complex $n \times n$ matrix. I wanna show that $ \mathbb I_n +A \bar{A}^t $ has an inverse. I tried find the actual inverse to no avail; then I looked at eigenvalues, also no luck. Any ...
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0answers
29 views

Elementwise normal to vector of unknowns and non-defined matrix multiplications

I wonder about statement (1). It is given that u is a column vector, A and B are constant, symmetric, square matrices of such size that the expressions on the left hand side of (1) are well defined. ...
0
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0answers
11 views

Question about notation matrix of partial derivatives where one component has two indices

lets suppose we have a vector ($\delta_{11}, \delta_{12}, \dots, \delta_{jk}$) where $\delta_{jk} = \alpha_j + \beta_k$, i.e., each element is build up of two components. The first index $j$ specifies ...
3
votes
1answer
43 views

Reference request for positive matrices

I would much appreciate someone suggest me a text book which covers stochastic matrices in depth with all relevant theories.Thanks
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1answer
45 views

How to inverse this matrix?

A long time ago I used to do it very fast and easily [A long time ago]. I would like to refresh my memory on what are the methods, steps to do in order to calculate an inverse of a matrix? Let say I ...
2
votes
0answers
54 views

Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
1
vote
3answers
96 views

Deducing the exact solution of a ODE

In page 53 of Arieh Iserles's A first course in the numerical analysis of differential equations, he presents the following ODE: $(\vec{y})'=\Gamma\cdot\vec{y}$, $\vec{y}(0)=\vec{y_0}$ Using the ...
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1answer
54 views

Calculating Eigenvalues is only

Assume that the following is used: $$ A = \begin{pmatrix} 0& 1&\\ 2& 3&\\ 4& 5&\\ 6& 7&\\ 8& 9& \end{pmatrix} $$ Then calculating the Coveriance ...
0
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0answers
19 views

Estimate vector sum with fewer vectors

Let $M$ be a $n \times 72$ matrix. Also let the $n \times 1$ vector $V$ represent the sum of all columns of $M$. How can I find a reduced set ($<< 72$) of columns of $M$ that best represents ...
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0answers
46 views

Prove that every proper principal submatrix of $\lambda I-A$ is nonsingular under certain assumptions

Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda ...
0
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1answer
22 views

Matrix addition and definiteness

Is strict/weak negative/positive definiteness/semidefiniteness of matrices preserved under matrix addition? I tried to do this for 2x2 matrix but even this wasn't easy. (I tried to use the principal ...
0
votes
1answer
147 views

Looking for a formula to calculate DCT/FFT frequencies when cropping a matrix/image.

Given: A is a matrix of dimensions W1 x H1 . Cropping: Few rows and/or few columns were deleted from matrix A. We got matrix B of dimensions W2 x H2. Not more than 5% of matrix A rows/columns ...
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1answer
35 views

Question on matrix

Suppose I have a vector field $F(x)=Ax$ where $A$ is a matrix. How can I express $Sx$ without $A$ (use $F$ instead)? Here $S=\dfrac{A+A^T}2$ is symmetric part of $A$.
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votes
1answer
26 views

Partial Differentiation, Vector Valued Function Derivatives

$A$ is an $m\times n$ matrix and $b$ is an $m \times 1$ column vector. Vector-valued $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is given by $f(x) = Ax + b$. Find the derivative, $f'(x)$. I was ...
1
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0answers
519 views

How to calculate Rotation Matrix in android from accelerometer and magnetometer sensor

I found the rotation matrix returned by SensorManager.getRotationMatrix from link: What's the best 3D angular co-ordinate system for working with smartphone apps The rotation matrix is: But I ...