Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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2
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1answer
62 views

Find all the $2 \times 2$ matrices $A$ satisfying $A^2 = 0$

Find all the $2 \times 2$ matrices $A$ satisfying $A^2 = 0$. So far I have this: But I don't know to proceed.
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0answers
38 views
+50

Gauss-Newton Method not converging for my function

I need to solve the optimization problem $$ \min_{x\in \mathbb{R}^{3}}f(x) $$ where the function $f$ is defined as follows: $$ f(x_{1}, x_{2}, x_{3}) = \frac{1}{2}\left[\left(2x_{1}-x_{2}x_{3}-1 \...
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3answers
42 views

Assume that A and B are square matrices, so that $AB = B^2A$. Prove $(AB)^2=B^6A^2$. [on hold]

I have no idea how to solve this or if I miss any properties,I will appreciate any kind of help and explanation. Excuse me for my broken english, is not my first language
1
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3answers
61 views

For the purposes of DFT, is “the” primitive root of unity $w_n = e^{ 2\pi i / n }$ or $w_n = e^{-2\pi i / n }$?

I'm working on the section of Strang's Linear Algebra and Its Applications 4e that discusses discrete Fourier transforms. To make the exercises easier, I wrote myself a Python script that generates ...
9
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2answers
57 views

$\operatorname{rank} A = \operatorname{rank} A^2$ if and only if $\lim_{\lambda \to 0} (A+\lambda I)^{-1}A$ exists

Let $A$ be any complex $n\times n $ matrix. Prove that $\operatorname{rank} A = \operatorname{rank} A^2$ if and only if $\lim_{\lambda \to 0} (A+\lambda I)^{-1}A$ exists. I am stuck on this problem, ...
3
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1answer
46 views

Which vectors are stretched the most in a transform?

I thought that it would be obvious that in a transform the eigenvectors are the ones that are stretched the most as those are the directions in which the matrix acts. But according to this short video ...
18
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3answers
8k views

Frobenius norm of product of matrix

The Frobenius norm of a $m \times n$ matrix $F$ is defined as $$\| F \|_F^2 := \sum_{i=1}^m\sum_{j=1}^n |f_{i,j}|^2$$ If I have $FG$, where $G$ is a $n \times p$ matrix, can we say the following? $$...
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0answers
44 views

Is my understanding of matrices correct?

Let $V = R^2$ be our vector space with the unit base vectors $J(1, 0), K(0, 1)$. We have the linear map, $$T: V \to V$$ $$T(v) = v'$$ We can rewrite $\forall v \in V$ as a linear combination of the ...
7
votes
2answers
446 views

Differentiating matrix exponential

I know that $$\frac{d}{dt}e^{At} = Ae^{At}$$ However, in one lecture, I found the following $$\frac{d}{dt}e^{A^Tt} = e^{A^Tt}A^T$$ The lecture is as follows How to show the second case? ...
0
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1answer
16 views

Direct proof for angular velocity from direction cosine matrix

I am trying to work through the math of what should be a relatively simple proof of a direct definition of the angular velocity matrix starting from the direction cosine matrix. The reference for ...
1
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1answer
61 views

Question regarding the similarity of an invertible matrix with its inverse .

Find the set $S$ of all possible $n×n$ invertible matrices $M$ such that $M$ is similar to $M^{-1}$ . My approach Actually, I was thinking about this problem when I came across a theorem stating ...
1
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1answer
64 views

How to show $\frac {\partial a^{T}X^{-1}b}{\partial X} = -\left( X^{-1}\right) ^{T}ab^{T}\left( X^{-1}\right) ^{T}$? [duplicate]

I am struggling with this proof where $X$ is $m \times n$ matrix, $a$ is $m$ vector, $b$ is $n$ vector. $$\frac {\partial a^{T}X^{-1}b}{\partial X} = -\left( X^{-1}\right) ^{T}ab^{T}\left( X^{-1}\...
0
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1answer
32 views

Derivative of log of multivariate Normal respect to Matrix?

I am wondering how to calculate the gradient of a multivariate Normal respect to L where $\Sigma=LL^{T}$. I know what would be the derivative respect to $\Sigma$ and I can use chain rule to get the ...
-1
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1answer
22 views

Rank of product of a matrix and its Moore-Penrose inverse [on hold]

Given a m x n matrix X with rank r: Does the product of X and its MP inverse always have rank r? If yes, why? Thanks
0
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1answer
37 views

Which one is correct between “$(Ax=0)$ $\land$ $(x\neq 0)$ $\iff det(A)=0$ ” and “$(Ax=0)$ $\land$ $(x\neq 0)$ $\implies det(A)=0$”?

As referred to Why non-trivial solution only if determinant is zero, that says "$(A−\lambda I)x=0$ has a nontrivial solution (a solution where $x\neq 0$) if and only if $\det(A−\lambda I)=0$ " which ...
7
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2answers
115 views
+50

Every matrix of the centralizer of the centralizer of a matrix is a polynomial in that matrix

Let $V=M(n,\mathbb C)$. For a subset $S \subseteq V$, let $C(S):=\{A \in V | AB=BA, \forall B \in S \}$ . How to prove that for every $A\in V$, we have $C(C (\{A\})) \subseteq \{ p(A) | p(t) \in \...
1
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0answers
25 views

Are the zero eigenvalues of a Laplacian matrix semi-simple?

It is known that the Laplacian matrix $\mathcal{L}$ for a directed weighted graph has at least one zero eigenvalue. If it has more than one zero eigenvalue, will there be non-trivial Jordan blocks ...
10
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3answers
4k views

How do I Generate Doubly-Stochastic Matrices Uniform Randomly?

A doubly-stochastic matrix is an $n\times n$ matrix $P$ such that $\displaystyle\sum_{i=1}^n{p_{ij}}=1$ and $\displaystyle\sum_{j=1}^n{p_{ij}}=1$ where $p_{ij}\ge 0$. Can someone suggest an ...
8
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1answer
2k views

What's the algorithm of finding the convex combination of permutation matrices for a doubly stochastic matrix?

According to Birkhoff, $n$-by-$n$ stochastic matrices form a convex polytope whose extreme points are precisely the permutation matrices. It implies that any doubly stochastic matrix can be written as ...
10
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1answer
1k views

Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
1
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1answer
93 views

What are the facets of the Tridiagonal Birkhoff $d$-polytope $\Omega^t_{d+1}$?

The Birkhoff $d$-polytope $\Omega_{d+1}$ is the convex polytope in $\mathbb{R}^{d+1}$ $\times$ $\mathbb{R}^{d+1}$ of doubly stochastic matrices: • All matrices contained in $\Omega_{d+1}$ have non-...
0
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1answer
57 views

Is there a representational face lattice for the tridiagonal Birkhoff polytope?

Is there a representational face lattice for the tridiagonal Birkhoff polytope of dimension d; i.e., $\Omega^t_{d+1}$? That is, is there a system of representing the faces of $\Omega^t_{d+1}$ so that ...
0
votes
1answer
19 views

Positive-Definite Matrix Question

I want to prove that the matrix is positive definite using the fact that: If $A$ is symmetric and $\langle x, Ax \rangle$ > $0$ for a nonzero vector $x$ then $A$ is positive. So I have the ...
3
votes
1answer
503 views

Projection onto Birkhoff Polytope

Suppose we would like to compute the Euclidean projection of an arbitrary matrix $A$ onto the Birkhoff polytope, the set of doubly-stochastic matrices. Under some conditions on $A$, Sinkhorn's ...
2
votes
1answer
23 views

Determine whether elements of a set are bounded above, below, or to the side of another in 3 or more directions

I'm having a difficult time expressing an algorithm mathematically. Assuming a 2D matrix of $x$ and $y$ elements, I have a set of pairs for elements that have been found and definitely exist ($A$), ...
0
votes
1answer
69 views

Do $A$ and $A^T A$ share an eigenvector?

I have been learning about singular value decomposition from http://www.ams.org/publicoutreach/feature-column/fcarc-svd and they say that orthongoal vectors in the domain are mapped to orthogonal ...
3
votes
1answer
33 views

Trace of matrix $A^{\ast}A$

Given a $n \times n$ matrix $A$ with complex entries. And $A^{\ast}$ represents the conjugate transpose of $A$.Then If $\left | tr{\left ( A^{\ast}A \right )}\right | <n^2$, then $\left |a_{...
-3
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1answer
39 views

(A−λI)x=0 and x≠0 iff det(A−λI)=0: Why [[1,1],[1,1]][[2],[3]] = [[5],[5]] ≠ 0 when det([[1,1],[1,1]]) = 0?

As refered to Why non-trivial solution only if determinant is zero, I wonder why \begin{gather} \begin{bmatrix} 1 & 1 \\1 & 1 \end{bmatrix} \begin{bmatrix} 2 \\3 \end{bmatrix} = \begin{...
0
votes
0answers
42 views

Boolean matrix factorization when a set of possible factors is known

Suppose I have a Boolean matrix $M$ which is a $m \times m$ square matrix. And there is a set of $m \times m$ Boolean matrices $M_1, M_2, M_3,\ldots, M_7$ such that $M$ is guaranteed to be a product ...
1
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1answer
46 views

Derivative of trace of a product containing an inverse matrix

What's the derivative of $$f(X)=\text{Tr}(YX^{-1})$$ with respect to $X$, where $X$ and $Y$ are square matrices of the same dimension? My first attempt is to apply the chain rule as: Let $h(X)=X^{-1}$...
1
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1answer
65 views

Existence of a symmetric matrix $X$ such that $XBX = A$

let $A,B$ be two positive semi-definite $n\times n$-matrices such that $$\mathrm{Range}(B^{1/2}AB^{1/2})=\mathrm{Range}(B)$$ and $$\mathrm{Rank}(A)=\mathrm{Rank}(B)=n-1$$ so is there a real ...
0
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0answers
19 views

Controllability Gramian via integration by parts

I'm trying to program something like this article. On page 3, there's the following equation (eq. 6) that should be expressible in closed-form: $$\int_0^te^{(A(t-t'))} M e^{(A^T(t-t'))}dt'$$ where $...
4
votes
2answers
83 views

Is there a geometric interpretation about the euclidean distance between of 2 matrices?

The Euclidean distance between points p and q is the length of the line segment connecting them ($\overline{\mathbf{p}\mathbf{q}}$). $$\begin{aligned}d(\mathbf {p} ,\mathbf {q} )=d(\mathbf {q} ,\...
0
votes
3answers
32 views

Finding n-th power of a 2*2 matrix with 2 identical eigen values

If$$ A = \begin{pmatrix} 3 & -4 \\ 1 & -1 \\ \end{pmatrix} $$ prove that $$A^k = \begin{pmatrix} 1+2k & -4k \\ k & 1-2k \\ \end{pmatrix}$$ Now the first ...
-1
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0answers
37 views

How to multiply vectors of a same matrix?

Let's, for example, consider the $3\times 6$ block matrix $$ \begin{bmatrix} 2&4&0 &\space &\space8&6&2\\ 5&9&0 &\space &\space 1&5&4 \\ 4&7&...
3
votes
3answers
222 views

Showing that the limit of non-eigenvector goes to infinity

Let $A$ be a $3$ by $3$ real matrix with the triple eigenvalue $1$. Also, further suppose its eigenspace corresponding to $1$ is only of dimension $1$. Thus, we can find a basis of $\mathbb{R}^3$, ...
2
votes
1answer
2k views

Least square solution based on the pseudoinverse solved efficiently with singular value decomposition

Hi apologies it's hard to type out the problem, I have a lecture slide on neural networks. It says the fitting error gives the matrix: N by M matrix of thi's multiplied by Mx1 weights minus Nx1 ...
1
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2answers
32 views

Given a Hermitian matrix $A$, prove that $(A-iI)$ is nonsingular

The exercise is to prove that, given $A$ a Hermitian matrix, then $(A-iI)$ is nonsingular. I tried to think about what it meant to be nonsingular, like $(A-iI)X=0$ have not only the trivial solution, ...
0
votes
1answer
38 views

Is it always possible to swap columns of a matrix by a left hand side multiplication?

I was thinking about swapping the columns of the matrix. It is well known that if you want to swap 2 columns of a matrix, you do a right hand side multiplication with a permutation matrix $T_{ij}$, ...
0
votes
0answers
9 views

Parameterizing rotation matrix

A general rotation matrix in terms of Euler angles is given by $$ \mathcal{R}=R_{\hat z}(\alpha)R_{\hat y}(\beta) R_{\hat z}(\gamma). $$ Working out the matrix multiplication we obtain the known ...
2
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0answers
21 views

trace of the product of two projection matrices [on hold]

If $P_{i}=\frac{\alpha_{i}\otimes\beta_{i}}{(\alpha_{i},\;\beta_{i})}=\frac{\alpha_{i}\beta_{i}^{T}}{\alpha_{i}^{T}\beta_{i}}$, where $P_{i}$ are rank-1 projection matrices and $(\alpha_{i},\;\beta_{i}...
7
votes
2answers
14k views

How to prove $(AB)^T=B^T A^T$

If $A$ is $m \times n$ and $B$ is $n \times p$ matrices, prove that $(AB)^T = B^T A^T$. Matrices' elements are $A = [a_{ij}], B = [b_{ij}]$. Let $C=AB=[c_{ij}]$, where $c_{ij} = \sum_{k=1}^n a_{ik}...
2
votes
4answers
63 views

Matrixes of higher order like $M_{\aleph\times \aleph}$ [on hold]

I was wondering if thinking about matrixes of the form $M_{\aleph\times \aleph}$ and assigning properties to them like matrix multiplication makes sense and useful in some way. note: $\aleph$ is the ...
1
vote
1answer
54 views

Identities $e^{A \otimes I} = e^{A} \otimes I$ and $e^{I \otimes A} = I \otimes e^A$

Suppose I have two square matrices of the same dimension, say $A_1, A_2$ over the complex numbers. The Kronecker sum is defined by $A_1 \oplus A_2 = A_1 \otimes I + I \otimes A_2$, and moreover $A_1 \...
0
votes
0answers
24 views

Block diagonal matrix as Kronecker product

Let $$X=\begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & ... & A_n \end{bmatrix}$$ Where the $...
7
votes
1answer
118 views

$\text{vec}\left(A\otimes B\right)$ is not $\text{vec}\left(A\right) \otimes \text{vec}\left(B\right)$

Let $A$ and $B$ be two square matrices of dimension $a$ and $b$. $\text{vec}\left(\cdot\right)$ is the vectorization of a matrix. Now $v_0=\text{vec}\left(A\otimes B\right)$ is not $v_1=\text{vec}\...
1
vote
1answer
42 views

Does the notion of a contraction operator depend on the norm which induces the metric?

Let $\lVert \cdot \rVert_a$ and $\lVert \cdot \rVert_b$ denote the $\ell_a$ and $\ell_b$ norms on $\mathbb{R}^n$ ($a,b$ are positive integers or $\infty$, $a \neq b$). Let $M_n(\mathbb{R})$ be the ...
0
votes
1answer
228 views

How the Wronskian works

To prove linear independence of a set of functions, we say that given their Wronskian matrix $~W~$, $~W_x = 0~$ implies trivial solution $~(0,0,0,\cdots)~$ if the value (determinant) of the Wronskian ...
2
votes
1answer
44 views

How to show the interpretability of NMF by a small qualitative example on a toy data?

In some paper, such as, Nonnegative Matrix Factorization: A Comprehensive Review, I see the interpretability of Nonnegative matrix factorization (NMF). However, I don't know the means of this. How to ...
2
votes
3answers
823 views

How do I determine the transformation matrix T of the coordinate transformation from the base E to the base B?

In $\mathbb{R}^3$ the canonical basis $E=\left (\mathbf{e_1},\mathbf{e_2},\mathbf{e_3} \right )$ and $B=\left (\mathbf{b_1},\mathbf{b_2},\mathbf{b_3} \right )$ with $\mathbf{b_1}=(1,2,4)^T$, $\...