All Questions
5,882 questions
7
votes
1
answer
919
views
Association scheme on injective functions
This problem arises while studying the complexity of algorithms and I am quite unfamiliar with the subject.
Consider the set F of injective functions from {1..N} to {1..M}
we can define an ...
- 12.1k
5
votes
6
answers
4k
views
Thorough Introduction to Singular Value Decomposition
Can you suggest a book that has a thorough introduction to Singular Value Decomposition?
CommunityBot
- 1
1
vote
1
answer
434
views
Tori acting on vector spaces
Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now ...
- 7,800
4
votes
1
answer
1k
views
Stability of Conjugate Gradient Method
When dealing with spd matrices of relatively low condition number, how likely is it (and is it easily provable) that the conjugate gradient method will always be able to find the solution without ...
- 20.2k
5
votes
2
answers
2k
views
Matrices whose exponential is stochastic
The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
- 60.5k
8
votes
3
answers
888
views
Techniques to bound products of upper triangular matrices and their inverses
Let $A_0, \dots, A_{n-1}$ be upper triangular matrices with ones on the diagonal. Let $B_{n-1}, \dots, B_0$ be of the same form.
I am interested in bounding
$$|| A_0 \dots A_{n-1} B_{n-1}^{-1} \dots ...
- 785
9
votes
3
answers
1k
views
Vector spaces with natural bases
Sergeib's question asks about vector spaces without a natural basis.
Actually, I would claim (apparently in accord with many comments and answers to Sergeib's question ) that this is the default ...
CommunityBot
- 1
3
votes
1
answer
456
views
Standard name for basis-independent submatrices?
Given a linear map $T:H\to H$ on an inner-product space $H$ and a subspace $K\subseteq H$, define the map $T_K = \pi_K T \pi_K^* :K \to K$, where $\pi_K:H\to K$ is the orthogonal projection.
As an ...
- 7,329
3
votes
1
answer
375
views
Connections between a polytope's symmetry group and the existence of periodic orbits
Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
- 1,675
6
votes
5
answers
1k
views
A signature inequality?
Given two real symmetric matrices $A$ and $B$ of common square size $n$ with no strictly negative eigenvalues, can the symmetric matrix $AB+BA$ have strictly more than $n/2$ eigenvalues which are ...
- 2,704
7
votes
5
answers
2k
views
Explicit invariants (under change of basis) of maps $V \to V \otimes V$.
It is standard to construct numbers associated to a linear transformation $f: V \to V$ of a finite-dimensional vector space which are invariant under change of basis. The coefficients of the ...
CommunityBot
- 1
-1
votes
2
answers
806
views
The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
- 156k
2
votes
0
answers
1k
views
Good sources for linear algebra for convex optimization and graph analysis?
What are some good sources for linear algebra for convex optimization and graph analysis?
In Particular, is Gilbert Strang's MIT course suitable, or some other online course? I prefer online courses (...
- 2,383
12
votes
2
answers
828
views
Matrices into path algebras
I was thinking about quivers recently, and the following idea came to me.
Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …,...
- 2,992
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 4,462
0
votes
2
answers
371
views
Quantum observables
Let H be a Hilbert space and A, B two non-commuting bounded linear operators. Let Com(A,B) be the set of bounded linear operators C which commute both with A and B.
Question 1 : What is known about ...
- 1,444
9
votes
1
answer
1k
views
(Elementary?) combinatorial identity expressing binomial coefficients as an alternating sum over permutations.
Background
I came up with this while trying to find a sort of high-level exposition of the exterior algebra of a vector space. Let $V$ be a vector space of dimension $n$ over $\mathbb{C}$, and let $...
- 118k
6
votes
2
answers
2k
views
Efficient approximation of a matrix and its inverse
Assume that $ A $ is a real $ n\times n $ matrix whose rows constitute an orthonormal basis of $ \mathbb R^n $.
Informal statement of question: Assume we want to approximate $ A $ by a rational ...
- 156k
7
votes
8
answers
1k
views
Ways to Synthesize Topics in Linear Algebra
Hello, I am currently studying linear algebra right now. In general, the material is pretty straight-forward but it doesn't seem particularly interesting. I suppose that the main thing that I am ...
- 1,882
2
votes
0
answers
4k
views
Eigenvalues of sum of commuting matrices [closed]
With reference to the following thread :
Eigenvalues of Matrix Sums
Answer by Jonas Meyer is as follows :
If 2 positive matrices commute, than each eigenvalue of the sum is a sum of eigenvalues of ...
CommunityBot
- 1
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(...
CommunityBot
- 1
1
vote
1
answer
736
views
Matrix Conjugates over Finite Fields
Thinking about Diffe-Hillman for matrices brought me to the following question.
Given $\mathbb{F}_{p^k}$ the finite field with $p^k$ elements when can we find non-trivial solutions to
$\begin{...
- 20.8k
3
votes
2
answers
375
views
Invariant subspaces of subalgebras of $M_n(C)$
Given a subalgebra E of $M_n$ (nxn complex valued matrices), what can we say about the subspaces F of $M_n$ such that $EF \subset F$? Googling for an answer gives me the reference:
Israel Gohberg, ...
- 18.7k
0
votes
2
answers
579
views
Linear algebra inequality
I'm wondering (hoping) if an inequality is true. Please can anyone help me?
Let $V$ be a complex vector space $dim_{\mathbb{C}}(V)=n$
with a hermitian scalar product $h$.
Let $v,a, b \in V$.
Is it ...
- 19.6k
3
votes
1
answer
632
views
What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors?
Construction
Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2:
$P^{(i)} = |a\rangle_i\langle a| + |...
CommunityBot
- 1
4
votes
1
answer
548
views
O(n^2) algorithm to approximate the sum of the log of the singular values of a matrix
Given an $M \times N$ matrix of rank $N$ ($M \ge N$) with $i^{th}$ singular value $\sigma_i$, does their exist an $O(M^2)$ algorithm for approximating the sum $ H =\sum_{i=1}^N \log(\sigma_i)$ with ...
- 114k
9
votes
2
answers
2k
views
Jordan Form Over a Polynomial Ring
Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (...
- 14.5k
1
vote
2
answers
917
views
Any known compact expression for
Is there any known compact expression for the sum
$$S_{k} = \sum_{i=1}^{k} A^{i-1} P Q^{k-i}$$
where $A$, $P$ and $Q$ are respectively $m \times m$, $m \times n$ and $n \times n$ matrices?.
You can ...
- 6,206
-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
11
votes
1
answer
688
views
Existence of a pair of matrices in SL(2,Z) satisfying certain constraints on the spectral radius
Some background: my coauthors and I are working on a problem which deals with the exponential growth rates of certain infinite products of matrices. One of the sub-problems which arises in this ...
- 68.9k
1
vote
1
answer
356
views
Finding the $J$ for a symplectic vector space
I found something strange when I was working on some other problems.
I thought the triple intersection description of the unitary group said that any two of $(g, \omega, J)$ determines the third ...
- 1,051
2
votes
0
answers
5k
views
A system of linear equations with linear constraints
Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\...
- 13.5k
8
votes
2
answers
3k
views
Centralizers in GL(n,p)
There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is ...
- 10.7k
14
votes
2
answers
937
views
Involutions in GL_n(Z)
Is there a classification of involutions in $\text{GL}_n(\mathbb{Z})$?
Here's some more details about what I mean. Consider $f \in \text{GL}_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an ...
- 14.5k
4
votes
0
answers
790
views
Is it possible to use linear programming to solve this problem?
I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...
- 41
2
votes
1
answer
455
views
A question on matrix decomposition.
Is the following claim true?
Claim Let $A, B\in C^{n\times n}$ with $rank(A)=rank(B)=r$. Then there exist nonsingular matrices $P_1, P_2, Q_1, Q_2$ such that
$$ Q_1AP_1=Q_2BP_2=\left(\begin{array}{...
CommunityBot
- 1
4
votes
2
answers
1k
views
Simultaneous Block decomposition of a set of orthogonal projections
An orthogonal projection is an Hermitian matrix $P$ such that $P^2=P$.
Denote $U^*$ the conjugate transpose of a matrix $U$.
It can be easily shown that for two projections $P_1$ and $P_2$, there ...
- 9,132
1
vote
1
answer
419
views
Is the direction of the longest line of a polytope unique?
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions.
The affine subspace is given by:
$X \mbox{ u} = y$
where
$u$ &...
- 13.6k
12
votes
0
answers
349
views
Matroids with prescribed independent sets
Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
- 1,791
3
votes
2
answers
2k
views
How can we explicitly find the maximum eigenvalue of a tridiagonal matrix?
I just came across a matrix of the form
$A:=\begin{pmatrix}
0&-\frac{c_0}{b_0}&0&\cdots&0\\-\frac{a_1}{b_1}&0&-\frac{c_1}{b_1}&\cdots&0\\0&-\frac{a_2}{b_2}&0&...
- 3,217
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 ...
- 8,491
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. ...
- 17.9k
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
13
votes
2
answers
8k
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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 ...
- 8,298
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 $\...
- 23.5k
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
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 $\...
- 10.7k
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 ...
CommunityBot
- 1
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 ...
- 156k
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 ...
- 4,776