All Questions
6,176 questions
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Which linear transformations between f.d. Hilbert spaces contract the inner product?
Given two finite-dimensional Hilbert spaces $U, V,$ a linear transformation $T:U\to V$ contracts the inner product if for all $x,y \in U,$
$$\langle x,y \rangle_U \ge \langle Tx, Ty\rangle_V.$$
...
- 10.3k
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votes
1
answer
385
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Adding a multiple of the Identity to a LU factorized matrix
Suppose a square, dense, symmetric matrix $A$ has been factorized into $L$ and $U$ components by performing a LU decomposition. Now let $B = A+\lambda I$. Is there any way to efficiently compute the ...
CommunityBot
- 1
1
vote
0
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396
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Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922
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votes
2
answers
410
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length of decompositions into elementary matrices
The Gaussian algorithm tells us, that for any field $k$ a $n\times n$-matrix over $k$ can written as a product of at most $C$ elementary matrices ($C\sim n^2$).
I am wondering, whether such a ...
- 25.5k
4
votes
3
answers
3k
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Making MATLAB svd robust to transpose operation
I'm playing with MATLAB's svd function to compute the svd of
[ 1 4 7 10
2 5 8 11
3 6 9 12 ]
When I type [U1, ~, ~] = svd(...
- 86
2
votes
1
answer
857
views
What is the relationship between singular value decomposition and solving linear systems?
It is known that solving systems of linear equations is reducible to SVD in a straightforward way; if you want to solve $\mathbf{Ax}=\mathbf{b}$, then you can perform SVD on $\mathbf{A}$ and minimize $...
- 2,019
3
votes
1
answer
591
views
Rank-1 decomposition conjecture for matrix with linear function elements
Can Anyone prove the following conjecture?
Consider $k$ rational function vectors $V_1(x_1,\cdots,x_n),\cdots,V_k(x_1,\cdots,x_n)$. They are called \textbf{linearly dependent} if there exists ...
- 52.3k
3
votes
3
answers
1k
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Structure theorem for finitely generated Z[G] modules
For a finite abelian group $G$ is there an analogue of structure theorem for finitely generated modules like for P.I.D. rings but with $Z[G]$ group ring over integers instead ?
- 57.6k
14
votes
2
answers
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What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
- 491
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3
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421
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Existence of tensor product of subalgebras
Let $\mathcal{G} = \mathbb{M}_n(\mathbb{C})$ be an $n$-by-$n$ matrix algebra over complex numbers. Next let $\mathcal{A} \cong \mathbb{M}_d(\mathbb{C})$ be a subalgebra of $\mathcal{G}$ and assume $d$ ...
- 20.8k
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3
answers
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Splitting the determinant polynomial into linear factors - a Dedekind problem
Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial
$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\...
- 52.3k
5
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4
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Diagonalization of Infinite Hermitian matrices
We know that $n\times n$ square Hermitian matrices can be diagonalized and have real eigenvalues.
Suppose I have a countable sized Hermitian matrix $A=(a_{ij})$ where the indices $i$ and $j$ run ...
- 60.5k
1
vote
1
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546
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A 3*3 matrix space problem
A matrix subspace $S\subset M_n(C)$ is called "good", if there is two linear independent elements of $S$, says $E_1,E_2$ which are simultaneously singular valued decomposable, i.e., $E_1=UD_1V$ and $...
- 4,705
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votes
2
answers
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Tight bound for sum of entries of the inverse of a nonnegative matrix
While playing around with certain non-negative matrices, I got stuck at the following question.
Let $A$ be a strictly positive-definite $n \times n$ matrix ($n \ge 3$), with ones on the diagonal, and ...
- 52.3k
9
votes
0
answers
1k
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coordinate-free proof of transitivity of norms or traces
Hello:
Suppose $A$ is a finite free $B$-algebra and $B$ is a finite free $C$-algebra. Does anyone
know a coordinate-free proof (i.e. without choosing bases) of the identity:
$N_{A/C} = N_{B/C}\circ ...
- 647
6
votes
1
answer
632
views
Norm of commutators (bis)
This question is slightly related to a popular one with the same title (see here).
Let $k$ be a field with characteristic zero. It is known (see Exercise 310) that a matrix $A\in M_n(k)$ is nilpotent ...
CommunityBot
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1
vote
0
answers
169
views
Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
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votes
3
answers
346
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Computational solutions to families of systems of linear equations
Question
Does there exist a computer package that will solve families of systems of linear equations over a field of prime characteristic?
An Example
Suppose I wanted to know when the following ...
- 20.2k
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 10.7k
1
vote
4
answers
946
views
Representation of Lie algebra sl_2.
Consider the Lie algebra $sl_2$
with the standard basis $(e,f,h),$ where
\begin{equation*}\label{sl2}
[h,e]=2\,e, [h,f]=-2\,f,[e,f]=h.
\end{equation*}
Let $V$ be finite-dimensional $sl_2$-...
- 1,335
4
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1
answer
314
views
For which linear endomorphisms can one find a basis such that the matrix is nonnegative?
Hi there,
Consider linear endomorphisms ("endos") of a finite dimensional vector space.
How can those endos be characterized, for which said vector space has a basis with respect to which the endo ...
CommunityBot
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1
vote
2
answers
6k
views
Square root of non-positive definite matrix
Finding square root of matrices using Cholesky decomposition is limited to positive definite matrices. Any other method to find square root of matrix which has some diagonal values approximately zero (...
6
votes
1
answer
301
views
Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 5,654
3
votes
1
answer
490
views
Bounds on operator 2-norms on partial traces of linearly related operators
Consider an arbitary positive semidefinite operator ρ, acting on ℂA ⊗ ℂB ⊗ ℂC, for A,B,C finite. Also, let P be an orthogonal projector on &#...
- 1,444
1
vote
2
answers
526
views
Eigenvalues and transpose
Please help me with the following question.
Let $F:\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a continuously differentiable mapping;
$F_{n}(x)$ be $n$-th iteration of $F(x)$, i.e. $F_{1}(x)=F(x)$, $F_{n}(x)...
- 3,928
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1
answer
4k
views
How complicated is infinite-dimensional "undergraduate linear algebra"?
The name "undergraduate linear algebra" in the title is a bit of a joke, and so I don't know how widely spread it is. To wit:
High school linear algebra is the theory of a finite-dimensional vector ...
- 56.6k
0
votes
1
answer
6k
views
Finding the determinant of a matrix with LU composition
Hi Mathoverflow
I hope you bear with me that my linear algebra knowledge is a little rusty, but I have a question that might potentially very easy to answer. Nevertheless it's been bugging me for a ...
- 3,283
2
votes
2
answers
369
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 56.6k
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1
answer
2k
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Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
- 16.1k
10
votes
3
answers
3k
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The largest eigenvalue of a "hyperbolic" matrix
Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\...
- 118k
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3
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498
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Morphisms between representations
I am looking at the automorphism group $G$ of a graph, represented as permutation matrices. The point in a proof I am trying to understand goes something like this:
"For any permutation matrix $P$ ...
- 1,327
2
votes
0
answers
104
views
Noisy bases for linear functions
For any $x \in \mathbb{R}^n$, the following statement is trivially true:
There exists a set $I \subset \mathbb{R}^n$ with $|I| \leq n$ such that for any $x' \in \mathbb{R}^n$, if $x \cdot y = x' \...
- 794
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1
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Determinant and symmetric power
Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
- 1,510
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2
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520
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Can a commutative, associative "multiplication" on an infinite-dimensional vector space be an isomorphism?
Let $V$ be a vector space (over $\mathbb C$, but I don't think it matters), and $m: V\otimes V \to V$ a "multiplication" that is associative and commutative (but I do not demand that it is unital). ...
- 53.3k
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4
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Why do we want to have orthogonal bases in decompositions?
In the decompositions I encountered so far, we all had orthogonal set of bases. For example in Singular Value Decomposition, we had orthogonal singular right and left vectors, in [discrete] cosine ...
- 12.7k
1
vote
2
answers
1k
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Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
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385
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Does knowing a conjugation of A to A^T determine eigenvalues of A?
Everybody knows that a square matrix $A$ has the same eigenvalues as
$A^T$. And it is clear that if $A^T=BAB^{-1}$ then $B$ maps eigenvectors
of $A$ to those of $A^T$. But I have not found any ...
- 2,713
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1
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How to resolve a wedge product of vector bundles
Let $X$ be an algebraic variety. Consider an exact sequence
$$0\to A\to B\to C\to 0$$
of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ (...
- 1,859
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2
answers
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Is the operator norm always attained on a $\{0,1\}$-vector?
Given an operator $f\colon R^m\to R^n$, can one always find a non-zero vector
$x\in \{ 0,1 \}^m$ such that $\|f(x)\|/\|x\|\ge0.01\|f\|$? (Here I denote by
$\|\cdot\|$ both the Euclidean norms in $R^m$ ...
- 56.6k
6
votes
3
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Pinching and positive definite matrices
A pinching over $M_n({\mathbb C})$ is an endomorphism $T$ where the $(i,j)$-entry of $T(M)$ is given either by $0$ or by $m_{ij}$, depending on the pair $(i,j)$. Let us say that a pinching is ...
- 9,389
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328
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What is this decomposition called?
Let $M$ be a positive semi-definite matrix, symmetric with real entries. Then $M$ can be written as $X X^T$. One way is by a Cholesky decomposition (unique for positive definite but not necessarily ...
- 173
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2
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Characteristic surface for systems of PDE
Despite the title, this is probably actually a question in linear algebra or algebraic geometry. Let me write the question(s) first, before I explain the background.
Problems
Let $h^{\mu\nu}_{ij}$ ...
- 52.3k
10
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7
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2k
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Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
CommunityBot
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votes
3
answers
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Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
- 43.2k
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3
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2k
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Torsion in GL_n(Z)
Fix some $n \geq 3$. It's hopeless to classify the torsion elements in $\text{GL}_n(\mathbb{Z})$, but I have a couple of less ambitious questions. It's well-known that for any odd prime $p$, the map ...
- 68.9k
-1
votes
1
answer
2k
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Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 16
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0
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608
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Orthogonal Projections in Lie Theory
I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
- 157
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1
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572
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When is a finite matrix a "good" approximate representation of an operator?
I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions
$\rho(r) = \sum_{i=1}^N q_i ...
CommunityBot
- 1
6
votes
3
answers
757
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How many products specify a sum?
Suppose that I have $n$ unknown variables $x_1,\ldots,x_n$. I wish to compute their sum:
$$Sum(x) = \sum_{i=1}^nx_i$$
However, the only access to these variables is through products: that is, for any ...
- 9,756
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1
answer
369
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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\...
CommunityBot
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