All Questions
6,262 questions
8
votes
2
answers
2k
views
Proof of a fact about traces
I'm following the open courseware content on Machine Learning from Stanford University. In the lecture notes, it is given that
$$\Delta_A \ tr(ABA^TC) = CAB + C^TAB^T$$
which I tried but couldn't ...
- 497
0
votes
1
answer
249
views
Going from individual elements back to to matrix/vector notation [closed]
Note: Moved to math.stackexchange.com. Sorry for the off-topic question!
[Context: I'm working through problem set one of Andrew Ng's Machine Learning course, question 2, trying to derive the Hessian ...
- 101
0
votes
2
answers
2k
views
decomposition of an orthogonal matrix
Hi,
I have a matrix : $W=I+U^TV$
$dim(W)=(D,D)$
$dim(U)=dim(V)=(N,D)$ with $N < < D$
I need it to be orthogonal ie $W^TW=I$
which gives me : $V^TU+U^TV+V^TUU^TV=0$
From that point, i ...
- 133
3
votes
1
answer
2k
views
Singular values of differences of square matrices
Suppose $A, B \in \mathbb{R}^{n \times n}$. Let $\sigma_1(A),\ldots,\sigma_n(A)$ be the singular values of $A$, and let $\sigma_1(B),\ldots,\sigma_n(B)$ be the singular values of $B$. If I know these ...
- 794
28
votes
4
answers
2k
views
Matrices: characterizing pairs $(AB, BA)$
Let $A$ be an $m\times n$-matrix and $B$ an $n \times m$-matrix over the same field. Consider the matrices $C=AB$ and $D=BA$. It is probably well known (and not difficult to show) that the only ...
- 6,919
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 19.6k
0
votes
1
answer
879
views
identity for matrices whose determinant is 1.
For any matrices $A$,$B$ in $SL(2,\mathbb{C})$, let $M(A,B)$ be the 4 by 4 matrix whose columns are matrices $I$,$A$,$B$,$AB$. Then it is not hard to verify that $det M(A,B)= 2- tr[A,B]$.
Is there an ...
- 159
2
votes
3
answers
285
views
is there any efficient way to compute the follow matrix equations easily
Let $A$ and $D$ are $n\times n$ diagnal matrices, and $B$ is an $n\times n$ orthogonal matrix. Is there any efficient way to compute the follow matrix equations easily?
$\sum_{i=0}^{k} A^i \cdot B^T \...
- 21
22
votes
1
answer
33k
views
vector to diagonal matrix [closed]
For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector.
Is there a simple way to write this transformation ...
- 247
4
votes
4
answers
596
views
Generalization of Jordan Decomposition for Several Commuting Operators
Recently I became curious about the following question:
Let $V$ be a finite dimensional vector space over $k$ and let $A_1, \cdots, A_n: V \rightarrow V$ be a set of commuting maps. Question: ...
- 607
13
votes
1
answer
329
views
Spectral properties of finite metric sets
Given a finite metric set $S=\{P_1,\dots,P_n\}$, one gets a real symmetric matrix $M=M(S)$
with rows and columns indexed by elements of $S$ by setting
$M_{i,j}=d(P_i,P_j)$.
It is easy to see that $M$...
- 17.9k
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
13
votes
7
answers
4k
views
Status of the Hadamard Circulant conjecture
The following feels like a community wiki question, so I do it here:
Recently we have heard of a new proof of the Circulant Hadamard conjecture of Ryser
(a long standing difficult conjecture):
...
7
votes
1
answer
785
views
AX=XB and the Cecioni--Frobenius theorem
The Frobenius--Cecioni theorem states that if $A$ and $B$ are square matrices with entries in a field $k$ then the dimension of the $k$ vector space of solutions of
$$
AX=XB
$$
is given by the sum
$$
\...
- 758
1
vote
1
answer
254
views
Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
user7807
5
votes
1
answer
683
views
Finitely generated algebra in which every element is annihilated by a non-zero polynomial
Let $K$ be a field, and $A$ a finitely generated associative algebra over $K$. We suppose that $A$ has a unit and that every element $x$ of $A$ is annihilated by a non-zero polynomial $P_x$ depending ...
- 663
-1
votes
1
answer
185
views
eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $
Let $A$, $B$ and $C$ be symmetric matrices.
What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I $?
- 461
8
votes
1
answer
1k
views
Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
- 11.1k
10
votes
2
answers
3k
views
Maximizing the Smallest Eigenvalue of a Diagonally Dominant Matrix
Assume that we have a full-rank diagonally dominant matrix $A$, all the diagonal elements of which are positive, all the non-diagonal elements are negative, and the sum of the absolute values of the ...
- 133
2
votes
1
answer
205
views
Do unitary bijections act invariantly on irreducible representations?
Let $\mathcal{A}$ be a $C^*$ algebra. Let $(\pi, \mathcal{H})$ be a faithful, irreducible, unitary, Hilbert space representation of $\mathcal{A}$; i.e., $\pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{...
- 657
3
votes
1
answer
376
views
On some type of conjugate of elements of SL(n,Z)
Let $A\in SL(n,\mathbb{Z})$ and $B\in\mathcal{M}_n(\mathbb{Z})$ s.t. $\det(B)\ne 0$. Is it possible to find a power of $B^{-1}AB$ in $SL(n,\mathbb{Z})$?
- 33
4
votes
1
answer
3k
views
Cauchy-like inequality for Kronecker (tensor) product
General question first: upper/lower bound a sum of Kronecker products by its components. More specifically,
how is $$
\Vert\sum_{\alpha}S_{\alpha}\otimes B_{\alpha}\Vert$$
bounded by the operator ...
- 731
6
votes
3
answers
2k
views
Is there a notation for the symmetric / antisymmetric subspaces of a tensor power that distinguishes them from the symmetric / exterior power?
Let $V$ be a finite-dimensional vector space over a field $k$, say of characteristic $0$. The symmetric group $S_n$ acts on the tensor power $V^{\otimes n}$ in the obvious way, and this action defines ...
- 118k
2
votes
1
answer
769
views
Is it possible to perform PCA on a multimensional array?
Normally you have a matrix n x p and apply PCA to it. But in the article I'm reading, the author considers that the matrix has points in it. So instead of being n x p, it'd be, say, n x p x 2. He then ...
- 21
2
votes
2
answers
308
views
Analogue of an orthogonal subspace in a noneuclidian normed space
This question is related to https://mathoverflow.net/questions/50600/an-existence-question-on-linear-map. If the answer to this question is yes, it would solve the abovementioned other MO question.
...
- 3,595
3
votes
0
answers
682
views
How to bound the second largest eigenvalue of a transition matrix of a non-irreducible Markov chain?
I have found several bounds (e.g., Cheeger, Poincare) for the case that the Markov chain is irreducible and reversible, however my Markov chain has one absorbing state. Any bound would be helpful, but ...
- 31
1
vote
1
answer
464
views
eigenvalues of A⊕B
Let $A_{n\times n}=(a_{ij}),B_{n\times n}=(b_{ij}) \in M_{n}(\mathbb{R})$, where $a_{ij},b_{ij} \in \lbrace 0,1\rbrace$. Boolean sum of $A,B$ denoted by $(A \oplus B)_{n\times n}=(a_{ij}\oplus b_{ij})$...
- 461
3
votes
3
answers
1k
views
a "reverse Hadamard inequality"
Is there an inequality of the form $|\det(A)| \geq F(v_1, \ldots, v_n)$ for a real $n\times n$-matrix $A$ with columns $v_i$, $F \geq 0$?
user19475
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
53
votes
9
answers
13k
views
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,306
1
vote
4
answers
1k
views
Prove: if a1,...,an are uniformly distributed unit vectors, then a1*a1'+...+an*an'=n/2*I
Hello everyone,
I have a very interesting question on orthogonal projection matrices. Intuitively it is quite straightforward to understand. But for me it is not easy to prove.
In $R^2$ space, $a_i$,...
4
votes
1
answer
1k
views
dominant eigenvector
Hi, everyone! Is there any efficient way to simplify the following tensor product
$X \otimes X + X^T \otimes X^T$, where $X$ is a square $n \times n$ matrix.
My goal is to efficiently compute the ...
- 41
3
votes
2
answers
1k
views
The space of probability measures and its intersection with hyperplanes in the space of measures
Let $X$ be some uncountable standard Borel space (e.g., the real line).
Let $D$ be the set of Borel probability measures on $X$.
Let $M$ be the set of signed Borel measures on $X$
Now let $p_1,...,p_N$...
user12713
1
vote
1
answer
149
views
Question on a relation between minors of a particular kind of matrix
Hi!
Perhaps it is an easy question but i don't figure out how to prove it.
Let $a_1,...,a_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer ...
- 1,727
23
votes
3
answers
2k
views
Which vector spaces are duals ?
Every finite-dimensional vector space is isomorphic to its dual.
However for an infinite-dimensional vector space $E$ over a field $K$ this is always false since its dual $E^\ast$ is a vector space ...
- 47.5k
53
votes
5
answers
5k
views
Does this formula have a rigorous meaning, or is it merely formal?
I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
- 15.3k
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
0
votes
1
answer
322
views
Sparse Principal Components Analysis: Any practical examples with fixed rank correlation matrix?
Consider the problem of sparse principal component analysis:
$$\max_{||{\bf x}||_0=k,||{\bf x}||_2=1} {\bf x}^T{\bf A}{\bf x}$$
where a $k$-sparse $n$-dim. unit vector that "maximizes variance" is to ...
- 449
8
votes
1
answer
248
views
Operator compression preserving lowest energy eigenspace.
I have a large ($10^6$ by $10^6$) sparse ($0.4$% nonzero) hermitian matrix $H$ arising from the discretization of an elliptic PDE. I would like to approximate $H$ with a smaller matrix $H'$ in such a ...
- 817
18
votes
1
answer
1k
views
Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
1
vote
2
answers
393
views
Could the Kunneth decomposition of a motif depend on the choice of $l$?
Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\...
- 16.9k
0
votes
1
answer
2k
views
Solving 5 eqns with 6 unknowns in a 2x3 contingency matrix, is there a unique solution? [closed]
Background
I have the following equations:
$$a+b+c=6$$
$$d+e+f=15$$
$$a+d=5$$
$$b+e=7$$
$$c+f=9$$
This is a 2x3 matrix $[a b c, d e f]$ where the marginal totals are fixed. In addition, all of the ...
- 225
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
7
votes
4
answers
2k
views
Is the componentwise square-root of a positive-definite matrix also pos.-def.?
Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and
$B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$.
Is $B$ positive-definite whenever $A$ is?
In other words:
$\...
5
votes
2
answers
2k
views
Lower Bound on the Cost of Solving Linear System
The cost of solving a linear system ("exactly") with Gauss Elimination and other similar methods with a few right hand side and where the matrix has no structure is $\mathcal{O}(N^3)$ where $N$ is the ...
user11000
2
votes
0
answers
695
views
Pole data of meromorphic matrix function
Let $T(z)$ be a meromorphic square matrix function, that is - a matrix whose entries are complex meromorphic function of one variable.
Recall that such a $T$ is said to have a right pole of order $r$ ...
- 1,214
3
votes
1
answer
288
views
Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?
I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} H^i(...
- 16.9k
1
vote
1
answer
2k
views
Pseudoinverse of columns of a matrix
First, some background:
I'm working on an implementation in C# of Lemke's algorithm (for solving linear complementarity problems) based on this Matlab implementation: http://ftp.cs.wisc.edu/math-prog/...
- 151
16
votes
2
answers
905
views
Eigenvalues of an "oblique diagonal" matrix
I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...
- 537
1
vote
1
answer
868
views
Is there an Error on pg. 17 of Tromba's "Teichmuller Theory in Riemannian Geometry"?
I'm pretty sure that this is a minor error, but I could use some help here. So the book I'm referring to in the title is this book (MR1164870).
On pg. 16-17, he is proving that the space of almost ...
- 245