All Questions
6,287 questions
1
vote
1
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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
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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 ...
- 23.3k
4
votes
1
answer
938
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
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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
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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
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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
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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
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
2
votes
0
answers
2k
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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
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
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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
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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
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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.
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3
votes
1
answer
1k
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problems of subspace of M_n(C)
let $M_n(c)$ denote the n times n matrices over the complex number field. $N$ be a subspace of
$M_n(C)$.
1 If there is no unitary lies in $N$, what is the maximum of the dimension of $N$ can be?
...
- 1,503
12
votes
4
answers
1k
views
Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 11.1k
3
votes
1
answer
1k
views
Matrix approximation
Let A be an $m\times n$ matrix and $k$ be an integer. Assume that $A$ is non-negative. We want to find a scalar $\epsilon$ and an $n\times n$ matrix $B$ such that $A\leq A(\epsilon I + B)$ (where $\...
- 145
1
vote
2
answers
440
views
A problem concerning two symmetric matrices
Let A , B denote two symmetric matrices of the same order n. and Spec(A)=X , Spec(B)=Y.
If Spec(A+B)=X $\cup$ Y , proof thar AB=0.
here Spec(A) means the set of the engevalues of A.
This is a ...
- 363
19
votes
4
answers
2k
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Problems concerning subspaces of $M_n(\mathbb{C})$
Let $M_n(\mathbb{C})$ denote the n times n matrices over the complex number field. N be a subspace of $M_n(\mathbb{C})$.
If all the matrices in N are non-invertible , what is the maximum the ...
- 363
6
votes
4
answers
17k
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Linear Regression Coefficients W/ X, Y swapped
Let's say I have a linear regression model of the form $ y = B_x x + I_x + \epsilon $, where $B_x$ is the beta coefficient of the $x$ term, $I_x$ is the intercept term and $\epsilon$ is additive, ...
- 159
4
votes
8
answers
3k
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Theory of cones
Hi all,
Can anyone point me to some references to the theory of finitely-generated cones in euclidean space? I'd like to know in particular if there is a notion of basis/dimension/linear dependence ...
0
votes
1
answer
8k
views
Product of Positive Matrices
Is the product of non-negative definite matrices also non-negative definite? If not, let A and B be non-negative definite matrices, is '$\operatorname{tr}(A^T B) \ge0$' ?
- 37
12
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2
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2k
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Non-degenerate multilinear forms
Is there a standard notion of non-degeneracy for multilinear forms?
My motivation is simple curiosity, by the way!
- 47.3k
7
votes
3
answers
2k
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Sarrus determinant rule: references, extensions
SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...
- 73
18
votes
3
answers
8k
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Number of invertible {0,1} real matrices?
This question is inspired from here, where it was asked what possible determinants an $n \times n$ matrix with entries in {0,1} can have over $\mathbb{R}$.
My question is: how many such matrices ...
- 32.1k
30
votes
12
answers
14k
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Why are tensors a generalization of scalars, vectors, and matrices?
Take two vector spaces $V$ and $W$ over a field $F$. One may form the tensor product $V\otimes W$ and it fulfills an universal property. Elements of $V\otimes W$ are called tensors and they are linear ...
18
votes
3
answers
6k
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Number of unique determinants for an NxN (0,1)-matrix
I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore will trivially have zero determinant. While it ...
- 359
5
votes
3
answers
781
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Acyclic quivers differing only in arrow directions: functorial isomorphism of representation categories?
Let $Q$ and $R$ be two acyclic quivers which differ only in the directions of their arrows (i. e., the underlying undirected graphs are the same).
1. Does there exist an isomorphism of additive ...
- 33.8k
9
votes
2
answers
3k
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What tensor product of chain complexes satisfies the usual universal property?
Recall that a chain complex is a (finite) diagram of the form
$$ V = \{ \dots \to V_3 \overset{d_3}\to V_2 \overset{d_2}\to V_1 \overset{d_1}\to V_0 \to 0 \} $$
where the $V_n$ are (finite-dimensional)...
- 54.6k
38
votes
6
answers
11k
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Is there a version of inclusion/exclusion for vector spaces?
I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...
- 549
4
votes
0
answers
97
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bounded homogeneous quartics
If Q is a real homogeneous quartic on $R^N$,
$Q(x) = \sum_{1 <= i,j,k,l <= N} Q_{ijkl} x_i x_j x_k x_l$
what is the condition on the (totally symmetric) coefficients $Q_{ijkl}$ for Q ...
- 362
4
votes
2
answers
236
views
Order of "one minus automorphism"
This is something I am stuck on (it might well be trivial- in which case this is an embarassing question):
Let V be a dimension r vector space over Fp, the field with p prime elements (I also care ...
- 22.1k
66
votes
3
answers
4k
views
Does linearization of categories reflect isomorphism?
Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
- 6,162
10
votes
4
answers
8k
views
Any reference on multilinear algebra [closed]
Do you know any good reference on multilinear algebra?
3
votes
1
answer
589
views
A question on star-congruence.
We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two ...
- 1,858