All Questions
6,260 questions
5
votes
1
answer
745
views
Decide how many non-negative solutions a set of multivariate quadratic equations have
Given a set of multivariate, quadratic, non-homogeneous equations, is there a way to decide how many non-negative roots it have?
Some explanations:
All the coefficients are real numbers.
The number ...
- 183
8
votes
3
answers
414
views
What can be said about pairs of matrices P,Q that satisfies $(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$ ?
Let $P,Q$ be $n$ by $n$ invertible matrices. Suppose further that $P$ and $Q$ satisfies the following equation :
$$(P^{-1})^T \circ P = (Q^{-1})^T \circ Q$$
where $\circ$ denotes the Hadamard matrix ...
- 2,154
3
votes
2
answers
1k
views
"Main" diagonal of a matrix
Hello!
I'm in search of some (possibly statistical) measure for matrices. I want to classify a square matrix as having the largest numbers running along the main diagonal or along the anitdiagonal. ...
- 33
2
votes
0
answers
156
views
A solver of a noisy system containing pairs of very similar linear equations, this is not about accurate solving of ill-cond. s.
Let there be a possibly overdetermined system AX = B, where B are some measured data, with a low noise level. To cancel out the measurement noise, and to allow for more unknowns, a large B is acquired,...
- 21
17
votes
4
answers
10k
views
Prime/undecomposable matrices
Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
- 2,855
9
votes
2
answers
2k
views
Classification of adjoint orbits for orthogonal and symplectic Lie algebras?
This might be standard, but I have not seen it before:
Let $K$ be an algebraically closed field (of characteristic 0 if necessary). Let $G$ be the orthogonal group ${\bf O}(m)$ or the symplectic ...
- 10.7k
2
votes
3
answers
806
views
An Linear Algebra Inequality
How to prove the following inequality:
Let $X$ and $Y$ be $n\times m$ matrices with real entries. Prove that
\begin{equation}
\det\left(XY^T\right)^2 \leq \det\left(XX^T\right)\det\left(YY^T\right) .
\...
- 31
13
votes
2
answers
8k
views
AC in group isomorphism between R and R^2
Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of ...
- 8,501
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
2
votes
1
answer
987
views
Surjectivity of bilinear forms.
It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ ...
- 4,015
12
votes
2
answers
2k
views
(Path) connected set of matrices?
Let $N \in \mathfrak{M}_n(\mathbb{C})$ nilpotent, such that there exists $X \in \mathfrak M_n(\mathbb{C})$ with $X^2=N$ (take for instance $n>2$ and $N(1,n)=1$; $N(i,j)=0$ otherwise).
Denote by $\...
- 2,829
5
votes
1
answer
3k
views
Finite subgroups of GL_n(C)
A finite groups of $\mathrm{GL}_n(\mathbb C)$ of exponent $m$ necessarily have order $C$ verifying $C\leqslant m^n$ and $n! m^n$ divides $C$, but this condition is not sufficient, for instance $\...
- 2,829
2
votes
3
answers
1k
views
Problem with the proof of a corollary of Schur's lemma
I'm reading the book 'A course in Modern Mathematical Physics' by
'Szekeres' and encountered a problem in interpreting the proof of the
following corollary of Schur's lemma.
The corollary and the ...
- 81
37
votes
3
answers
5k
views
Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?
If $k$ is a characteristic $p$ field containing a subfield with $p^2$ elements (e.g., an algebraic closure of $\mathbb{F}_p$), then the number of isomorphism classes of supersingular elliptic curves ...
- 45.7k
2
votes
3
answers
2k
views
free Z-modules: Bases etc.
I need a reference which states which of the "normal properties of vector spaces" carry over to free $\mathbb{Z}$-modules.
Especially I am interested in things like: If you have a linear map between ...
- 1,282
21
votes
9
answers
19k
views
What is the best algorithm to find the smallest nonzero Eigenvalue of a symmetric matrix?
see title.
An algorithm is 'good' if it is able to distinguish between zero Eigenvalues and nonzero Eigenvalues.
- 979
2
votes
2
answers
3k
views
Multiple outliers for two variable linear regression
Problem
Visually, the "extreme" outliers in the following graph are somewhat obvious:
Question
Given:
T - Set of all temperatures
Y - Set of all years
ΣT - Sum of temperatures.
ΣY - Sum of years.
...
- 43
8
votes
4
answers
7k
views
Positive solutions of linear Diophantine equations
Let $A$ be a non-negative integer $k\times n$-matrix (i.e. each entry is non-negative and integer) with $rank(A) = k < n$. Let $b$ be a $k$-dimensional vector with positive integer entries. ...
- 351
1
vote
1
answer
369
views
A matrix with trace entries.
This question is related to On a positivity of a matrix with trace entries.
Let $A_1, \cdots, A_m$ be strictly contractive $n\times n$ complex matrices .Is it true that
$$\left(\begin{array}{cccc}Tr\...
- 1,858
13
votes
2
answers
1k
views
Combinatorial proof of (a special case of) the spectral theorem?
The spectral theorem for a real $n \times n$ symmetric matrix $A$ says that $A$ is diagonalizable with all eigenvalues real. If $A$ happens to have non-negative integer entries, it can be interpreted ...
- 118k
22
votes
1
answer
13k
views
Non-diagonalizable complex symmetric matrix
This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.
Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with ...
- 23.4k
4
votes
1
answer
940
views
Random projection and finite fields
Suppose we have, say, $n$ $2n$-dimensional linearly independent vectors over $\mathbb{F}_2$. We do a projection on a random $d$-dimensional subspace. We are interested in probability that images of ...
- 1,791
5
votes
1
answer
941
views
What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )?
I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language ...
- 10.7k
7
votes
1
answer
347
views
Nonexistence of determinantal functional equation for $\arccos$
Suppose I have distinct real numbers $a_i \in [-1,1]$, $i \in [k]$. I want to choose real numbers $b_j, j\in [k]$ such that the matrix $(\arccos(a_i b_j))_{i,j \in [k]}$ is nonsingular.
Is this ...
- 1,021
68
votes
4
answers
9k
views
explicit big linearly independent sets
In the following, I use the word "explicit" in the following sense: No choices of bases (of vector spaces or field extensions), non-principal ultrafilters or alike which exist only by Zorn's Lemma (or ...
- 63.1k
-3
votes
1
answer
3k
views
Are there infinitely many equivalence classes of similar matrices? [closed]
It is easy to show that similarity in matrices is an equivalence relation (two matrices A and B of same size being similar if there exists a matrix P such that B = PAP^(-1) )
Moreover, given a matrix, ...
- 2,855
14
votes
3
answers
872
views
How can we realize different combinatorial objects as the dimension of a construction on vector spaces? Are the resulting algebras useful?
Fix a vector space $V$ of dimension $n$ over some field $F$. Here are three commonly seen constructions:
its $k$th tensor power, $T^kV$, which has dimension $n^k$
its $k$th exterior power, $\Lambda^k(...
- 6,792
0
votes
1
answer
406
views
Operation of GL_n(Z/bZ) [closed]
I want to show, that $GL_n(\mathbb{Z}/b\mathbb{Z})$ operates transitively on
$X = \{ (v_1, \ldots, v_n) \in (\mathbb{Z}/b\mathbb{Z})^n \ | \ v_1\mathbb{Z}/b\mathbb{Z} + \ldots + v_n\mathbb{Z}/b\...
- 3
9
votes
1
answer
439
views
Connected subset of matrices ?
Let $m,n$ be positive integers with $m \leqslant n$, and denote by $\mu_M$ the minimal polynomial of a matrix.
Do we know for which $m$ the set $E_m$ of $M \in \mathfrak{M}_n(\mathbb{R})$ such that $\...
- 2,829
5
votes
0
answers
482
views
A class of determinants associated to Catalan-like Hankel determinants
The following matrices are related to some Catalan-like Hankel matrices. My question is whether direct computations of determinants of such matrices (i.e. without recourse to Hankel determinants) ...
- 5,622
0
votes
1
answer
508
views
Is this trace inequality true?
In comparing the norm of two operators, I come across the following problem.
Let $S\in M_{n}(\mathbb{R})$ be a symmetric matrix. $D_1=diag(\alpha_1,\cdots,\alpha_n)$, $D_2=diag(\beta_1,\cdots,\beta_n)...
- 1,858
9
votes
1
answer
1k
views
How to write down the determinant of a quasi-isomorphism?
This question about the determinant of a perfect complex reminded me of an old question that I had.
The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...
- 3,284
1
vote
0
answers
1k
views
Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
30
votes
14
answers
13k
views
Geometrical meaning of Grassmann algebra
I don't understand wedge product and Grassmann algebra. However, I heard that these concepts are obvious when you understand the geometrical intuition behind them. Can you give this geometrical ...
- 303
10
votes
2
answers
2k
views
Largest rank submatrix of a skew symmetric matrix
Is the following statement true?
Given a skew symmetric matrix M, among all of its largest rank sub-matrix, there must be one that is the principal submatrix of M.
- 103
4
votes
0
answers
352
views
"Cholesky decomposition" X=YY* for p-adic matrices?
Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
- 932
10
votes
1
answer
1k
views
Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
- 52.9k
12
votes
4
answers
752
views
Additive commutators and trace over a PID
I would like to find an example of principal ideal domain $R$, such that there exists a square matrix $A\in \mathfrak{M}_n(R)$ with zero trace that is not a commutator (i.e. for all $B,C \in \mathfrak{...
- 2,829
23
votes
1
answer
2k
views
Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf
Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would ...
- 50.8k
2
votes
0
answers
2k
views
A square matrix is congruent to its transpose
we know a square matrix is similar to its transpose, this result holds true over any field.
for they have the same "invariant factors".
Similarly, it has been proven that a square matrix is congruent ...
- 363
6
votes
2
answers
1k
views
Linear algebra and regular orbits
If $A$ is an $n\times n$ matrix over a field, and $A^{k} = I$, with $k$ the least positive integer such that this occurs, then must there be some vector $v$ such that $\{v,Av,A^{2}v,\dots,A^{k-1}v\}$ ...
- 10.7k
8
votes
1
answer
811
views
(0,1)-matrix congruence: is it known?
[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]]
By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...
- 4,229
6
votes
1
answer
3k
views
Is there good intution of the trace map?
I have never understood the trace map,not even after reading Geometric Interpretation of Trace. The problem with many answers in the above discussion is the geometric intuition does not apply to other ...
- 2,824
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
8
votes
2
answers
2k
views
A question on a trace inequality
Let $A, B\in M_{n}(\mathbb{R})$ be symmetric positive definite matrices. It is easy to see $Tr(A^2+AB^2A)=Tr(A^2+BA^2B)$. Numerical experiments indicate $$Tr[(A^2+AB^2A)^{-1}]\ge Tr[(A^2+BA^2B)^{-1}],~...
- 1,858
9
votes
1
answer
591
views
Waring's problem for matrices
Probably a well-know question, but I haven't solved it, so I'll ask.
I can show that every matrix in $M_2(\mathbb{R})$ is the sum of two squares of matrices in $M_2(\mathbb{R})$.
If $n>2$, I can ...
- 2,829
8
votes
2
answers
2k
views
Characterizing invertible matrices with {0,1} entries
Related to the question link text I was asking myself some time ago the following. Can one precisely describe the invertible n\times n matrices with{0, 1} entries? For example, is anything special ...
-2
votes
6
answers
3k
views
Is this an if-and-only-if definition of affine? [closed]
x -> A x+ b.
Quoted from Affine transformation:
In general, an affine transformation
is composed of linear transformations
(rotation, scaling or shear) and a
...
- 21
2
votes
3
answers
772
views
Matrix decomposition problem
Given a pair of distributions $x,y\in(0,1]^{n\times 1}$, so that $1^Tx=1$ and $1^Ty=1$,
I want to build a matrix $C$ (change matrix) that satisfy at least the following properties:
i) $C$ is ...
- 51
1
vote
2
answers
2k
views
Rank of ABA where B is positive definite
I have a n-by-k matrix A and a n-by-n matrix B, where B is positive definite. I can form the matrix $M = A^t B A$. Playing around, I always found $rk(M) = rk(A)$ but I can't prove this.
- 187