All Questions
5,883 questions
16
votes
3
answers
2k
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Hom(A,C) ⊗ Hom(B,D) injects into Hom(A⊗B,C⊗D): when? why?
Sorry for asking a linear algebra question on a research forum, but this seems to be either a case of extreme blindness on my side, or a case of a result lying much deeper than it seems.
The ...
- 33.8k
16
votes
3
answers
3k
views
A riddle about zeros, ones and minus-ones
I was asked this years ago, but I don't remember by whom, and have never managed to solve it.
Consider the $2^n \times n$ matrix of all vectors in {-1,1}$^n$.
Someone comes and maliciously replaces ...
- 435
16
votes
1
answer
3k
views
How to construct a basis for the dual space of an infinite dimensional vector space?
Let $V$ be an infinite-dimensional vector space over a field $K$. Then it is known that $\dim V < \dim V^*$. More precisely, by a result attributed to Kaplansky and Erdos, we have $\dim V^* = |K|^{\...
- 2,821
16
votes
3
answers
2k
views
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 73
16
votes
1
answer
3k
views
A property that forces the NORM to be induced by an INNER PRODUCT
Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$,
$$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$
I want to show that the norm is induced by an inner product. Any ...
user36763
16
votes
1
answer
1k
views
Is the theory of vector bundles just linear algebra done in a suitable topos?
Sheaves of sets on a space are somehow "parametrized sets". This is the philosophy by which one can do mathematics internal to a sheaf topos (of which theory I admit I know essentially nothing), with ...
- 23.3k
16
votes
2
answers
2k
views
Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$
The setup is as in this question:
Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}...
- 13.4k
16
votes
1
answer
3k
views
positive not completely positive maps
In extension to this question
Positive but not completely positive?
I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (...
- 3,873
16
votes
1
answer
711
views
A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space
While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
Question: Do there exist a non-trivial ...
- 1,101
16
votes
3
answers
1k
views
symmetric integer matrices
Suppose I have a symmetric positive definite matrix $M$ with integer entries. I want to decide whether $M = A A^t,$ with $A$ likewise integral. I assume that decision problem is NP-complete, as is the ...
- 96.4k
16
votes
2
answers
9k
views
Solving a quadratic matrix equation
This might be a well-known problem but I am having trouble to find this. For square matrices $X, A, B,$ how to obtain the general solution for $X$, for the quadratic matrix equation $X A X^{T} = B$ ? ...
- 1,538
16
votes
6
answers
13k
views
Showing block diagonal structure of matrix by reordering
Suppose we have a block-diagonal matrix $M$, but the block diagonal structure is not immediately apparent from looking at the matrix because the rows/columns are shuffled.
I wish to find a reordering ...
- 911
16
votes
2
answers
1k
views
Definitions of determinant by unique features
A well-known definition of the determinant is:
The determinant is the only function of a vector space of dimension $n$ to its underlying field which is multilinear, alternating and normalized.
See e....
16
votes
4
answers
782
views
$\sup \left\| A x + B y\right\|_2$ subject to $\left\|x\right\|_2 = \left\|y\right\|_2 = 1$
I'm interested in
$$\sup_{x, y} \left\| A x + B y\right\|_2$$ subject to
$$\left\|x\right\|_2 = \left\|y\right\|_2 = 1$$
where $A$, $B$ and $x$, $y$ are real matrices and vectors, respectively, of ...
- 1,667
16
votes
3
answers
15k
views
Interesting relationships between Cholesky decomposition and diagonalization
Let $\Sigma$ be a hermitian positive definite matrix and $L$ be its Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ is ...
- 1,902
16
votes
1
answer
851
views
Are primes linearly separable?
Let $X_1,\cdots,X_n$ be finite subsets of some set $Z$. Then the symmetric difference metric space:
$$d(X_i,X_j) = \sqrt{ |X_i|+|X_j|-2|X_i\cap X_j|}$$
can be embedded in Euclidean space. The value $|...
user6671
16
votes
1
answer
818
views
Lift chain complex from $\mathbb{F}_2$ to $\mathbb{Z}$
We start with a finite dimensional chain complex over $\mathbb{F}_2$, equipped with a basis. That is, we have finitely many finite dimensional $\mathbb{F}_2$-vector spaces $C_0,\dots,C_k$ with bases $...
- 518
16
votes
1
answer
774
views
Minimizing the excursion of a sum of unit vectors
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...
- 151k
16
votes
4
answers
930
views
Integer matrices whose determinant equals their norm
Let $M$ be an $2 \times 2$ matrix, with all entries in $\mathbb{N}$:
$$
M=
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} \;.
$$
So
$$
\mathrm{det}(M) = a d - b c \; .
$$
The
Euclidean norm
(...
- 151k
16
votes
3
answers
791
views
Random products of projections: bounds on convergence rate?
The von Neumann-Halperin [vN,H] theorem shows that iterating a fixed product of projection operators converges to the projector onto the intersection subspace of the individual projectors. A good ...
- 181
16
votes
1
answer
537
views
Balls in Hilbert space
I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
- 2,049
16
votes
1
answer
2k
views
Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
- 839
16
votes
1
answer
2k
views
Moore-Penrose Inverse as an adjoint
A Moore-Penrose pseudoinverse of a morphism $f: V \rightarrow W$ between Euclidean vector spaces is a map $g: W \rightarrow V$ in the other direction satisfying the identities
$fgf = f$
$gfg = g$
$(...
- 1,704
16
votes
1
answer
2k
views
Overlapping Gershgorin disks
We all know Gershgorin's Circle Theorem, which I will summarise for convenience. Let $A=(a_{ij})$ be an $n\times n$ complex matrix. Define the disks $D_1,\ldots,D_n$ by $$D_i = \Bigl\{ z : |z-a_{ii}|\...
- 37.7k
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
16
votes
1
answer
897
views
Hankel determinants of binomial coefficients
For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & ...
- 2,188
16
votes
1
answer
2k
views
Playing an (invertible) matrix game with two players
Players $A$ and $B$ take an empty $n \times n$ matrix and place, one by one, an element (say, a rational number) in an unoccupied place of this matrix. Player $A$ starts. The game ends if there is no ...
- 161
16
votes
0
answers
755
views
Is there a "natural" proof of the equality $4^2=2^4$?
This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
- 18.4k
16
votes
0
answers
574
views
Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?
It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
- 1,206
16
votes
0
answers
488
views
An inequality for matrix norms
Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically:
Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
- 4,484
16
votes
0
answers
809
views
Determinant inequality involving Hermitian, positive definite matrices
Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.
Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$
This question has been ...
- 269
16
votes
0
answers
784
views
How to explain the picturesque patterns in François Brunault's matrix?
How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...
- 19.6k
15
votes
5
answers
18k
views
Proving "almost all matrices over C are diagonalizable".
This is an elementary question, but a little subtle so I hope it is suitable for MO.
Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$.
The characteristic polynomial $T - \lambda I$ splits ...
- 7,442
15
votes
3
answers
4k
views
Non-diagonalizable doubly stochastic matrices
Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
- 731
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
- 151
15
votes
4
answers
1k
views
Determining if some permutation of a vector satisfies a system of linear equations
Let $A$ be a matrix and $x$ a fixed vector. How can we determine whether or not there exists a permutation matrix $P$ such that $APx=0$? Does this problem reduce to anything well-understood?
- 623
15
votes
3
answers
1k
views
Are automorphisms of matrix algebras necessarily determinant preservers?
Is every automorphism $\phi : A \to A$ of a subalgebra $A \subseteq M_n$ necessarily a determinant preserver?
I would assume that the answer is no in general, but I'm unable to find an example (or any ...
- 261
15
votes
3
answers
738
views
Finding lots of discrete vectors in fairly general position
How many vectors can there be in $\mathbb{F}_2^{2n}$ such that no $n$ of them form a linearly dependent set? The bounds I have so far are embarrassingly far apart, though that probably means I should ...
- 29k
15
votes
3
answers
1k
views
Group of matrices in which every matrix is similar to unitary
$\DeclareMathOperator\GL{GL}$Let $G$ be a subgroup of $\GL_n(\mathbb{C})$ such that for every $g \in G$ there exists $c \in \GL_n(\mathbb{C})$ for which $cgc^{-1}$ is unitary (or, which is the same, $...
15
votes
3
answers
24k
views
How to solve this quadratic matrix equation?
I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...
- 395
15
votes
3
answers
3k
views
Determinant of a $k \times k$ block matrix
Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...
- 301
15
votes
4
answers
2k
views
More than $n$ approximately orthonormal vectors in $R^n$
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well established that ...
- 1,160
15
votes
2
answers
997
views
Matrix equation $XAXBXC=I$
Let $A,B,C$ be unitary matrices. Does there always exist a unitary matrix $X$ such that $$(XA)(XB)(XC)=I,$$ where $I$ is the identity matrix? The quadratic equation $(XA)(XB)=I$ has the solution $A^*(...
- 213
15
votes
3
answers
1k
views
The normalizer of $\mathrm{GL}(n,\mathbf Z)$ in $\mathrm{GL}(n,\mathbf Q)$
It seems that the normalizer of $H=\mathrm{GL}(n,\mathbf Z)$ in $G=\mathrm{GL}(n,\mathbf Q)$ is "almost" equal to itself, that is,
$$
N_G(\mathrm{GL}(n,\mathbf Z))=Z(G) \cdot \mathrm{GL}(n,\mathbf Z)
...
- 303
15
votes
2
answers
1k
views
Positive quadratic polynomial
Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$.
Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$.
Is it possible to find a polynomial $\tilde q$ ...
- 45k
15
votes
4
answers
869
views
What is known about ordinary character values at involutions?
Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...
- 1,755
15
votes
3
answers
18k
views
angle between subspaces
Let $E$ be a finite dimensional real inner product space. I want to define the angle between two subspaces $E_1$ and $E_2$. This has a fairly obvious meaning if $E_1$ is 1-diemsnional: Take the ...
- 421
15
votes
2
answers
7k
views
Efficient rank-two updates of an eigenvalue decomposition (or more generally SVD)
Let $A$ be a symmetric matrix with eigenvalue decomposition $UDU^T$. Golub, et al.1 and Bunch, et al.2 have shown that given such an $A$, the eigenvalue decomposition of $A+\rho xx^t$ may be computed ...
- 947
15
votes
1
answer
1k
views
Existence of double eigenvalue
Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent.
Must there exist a nonzero real linear combination $aA + bB$ which has a repeated ...
- 42.8k
15
votes
3
answers
6k
views
Simultaneous diagonalization
I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ ...
- 5,243