All Questions
644 questions
1
vote
1
answer
223
views
Worpitzky-like identities?
Let $$r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$$
Computations suggest that $$r_{2k}(x)=\sum_{j=0}^{(k-1)^2}a(2k,j)\binom{k^2+x-j}{k^2}$$ and $$r_{2k+1}(x)=\sum_{j=0}^{k^2-k}a(2k+1,j)\...
- 5,622
1
vote
1
answer
276
views
Spectral decomposition of a $4\times4$ real nonsymmetric matrix with unknown elements
I'm trying to eigendecompose the following matrix $A$, i.e. to find $Q$ and $\Lambda$ such that
$$
A = \begin{bmatrix}
-\alpha & \alpha & -\gamma^{-1} & 0\\
\beta &...
- 207
15
votes
2
answers
473
views
Generalizations of summation methods of divergence series
If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
11
votes
1
answer
627
views
One question on circulant $\pm1$ matrices
Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property
$$AA^T=(n-1)I+J$$
where $I$ is the $n \times n$ identity matrix and $J$ ...
- 696
115
votes
3
answers
5k
views
The number $\pi$ and summation by $SL(2,\mathbb Z)$
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$. (it is the defect in the triangle inequality)
Then, we discovered by heuristic arguments and then verified by computer that
$$\...
- 5,053
66
votes
3
answers
4k
views
Does linearization of categories reflect isomorphism?
Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
- 6,162
66
votes
2
answers
8k
views
Geometric interpretation of characteristic polynomial
The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
- 2,071
63
votes
7
answers
9k
views
How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...
- 845
56
votes
21
answers
18k
views
Wonderful applications of the Vandermonde determinant
This semester I am assisting my mentor teaching a first-year undergraduate course on linear algebra in Peking University, China. And now we have come to the famous Vandermonde determinant, which has ...
53
votes
5
answers
5k
views
Does this formula have a rigorous meaning, or is it merely formal?
I hope this problem is not considered too "elementary" for MO. It concerns a formula that I have always found fascinating. For, at first glance, it appears completely "obvious", while on closer ...
- 15.3k
46
votes
2
answers
7k
views
Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
- 573
44
votes
2
answers
2k
views
Is this lemma in elementary linear algebra new?
Is anyone familiar with the following, or anything close to it?
Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces
over an infinite field $k$, and $V$ a subspace of $A\otimes_k B$
...
- 595
42
votes
4
answers
33k
views
What is the intuition for the trace norm (nuclear norm)?
I will word this question in terms of linear operators acting on $\mathbb{C}^n$ for simplicity. Feel free to provide an answer in terms of more general Hilbert spaces if you think it makes more sense ...
- 539
40
votes
6
answers
6k
views
Linear transformation that preserves the determinant
It seems "common knowledge" that the following holds:
Let $T$ be a linear transformation on $n\times n$ matrices with complex coefficients that preserves the determinant. Then there exists ...
- 511
38
votes
2
answers
6k
views
Over which fields are symmetric matrices diagonalizable ?
The question is motivated by this one real symmetric matrix has real eigenvalues - elementary proof:
Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed fields) ...
- 567
38
votes
10
answers
18k
views
Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
36
votes
4
answers
2k
views
Determinant of the random matrix $X^2+Y^2$
$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$.
i) When $n=2,3,4$, one ...
- 3,741
35
votes
3
answers
4k
views
A curious determinantal inequality
In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...
- 1,748
35
votes
4
answers
2k
views
Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?
A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...
- 2,527
35
votes
5
answers
13k
views
Linearity of the inner product using the parallelogram law
A norm on a vector space comes from an inner product if and only if it satisfies the parallelogram law. Given such a norm, one can reconstruct the inner product via the formula:
$2\langle u,v\rangle ...
- 26.8k
34
votes
3
answers
22k
views
Singular values of matrix sums
This is a follow-up question to this one about eigenvalues of matrix sums. Suppose you have matrices $A$ and $B$, and know their singular values. What can you say about the singular values of $A+B$?
...
- 6,342
34
votes
3
answers
3k
views
Quickly determining if a matrix has any PSD completion
Given $m$ entries of an $n \times n$ matrix, is it possible to determine in $O(m n)$ time whether there is any positive semidefinite completion?
Slightly more precisely: for simplicity let's assume ...
- 777
34
votes
3
answers
6k
views
Why is uncomputability of the spectral decomposition not a problem?
Below, we compute with exact real numbers using a realistic / conservative model of computability like Type Two Effectivity.
Assume that there is an algorithm that, given a symmetric real matrix $M$, ...
- 4,943
34
votes
13
answers
6k
views
Elementary applications of linear algebra over finite fields
I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...
33
votes
4
answers
10k
views
Definition of inner product for vector spaces over arbitrary fields
Is there a canonical definition of the concept of inner products for vector spaces over arbitrary fields, i.e. other fields than $\mathbb R$ or $\mathbb C$?
- 341
31
votes
1
answer
4k
views
Determinants of binary matrices
I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
- 12.7k
30
votes
7
answers
4k
views
When is a monic integer polynomial the characteristic polynomial of a non-negative integer matrix?
Suppose $P(x)$ is a monic integer polynomial with roots $r_1, ... r_n$ such that $p_k = r_1^k + ... + r_n^k$ is a non-negative integer for all positive integers $k$. Is $P(x)$ necessarily the ...
- 118k
27
votes
2
answers
1k
views
Uniquely reconstruct a matrix $M$ from its inverse $M^{-1}$ if $n$ elements of $M^{-1}$ are unknown and $n$ elements of $M$ are given
This question was motivated by a recent MO post. You know $n$ elements of the $N\times N$ matrix $M$ and you do not know $n$ elements of the inverse $M^{-1}$ (but you know the other $N^2-n$ elements ...
- 188k
27
votes
7
answers
9k
views
Why are two "random" vectors in $\mathbb R^n$ approximately orthogonal for large $n$?
I saw that two random independent vectors are approximately orthogonal in high dimensional space.
How can I prove this?
And is there an intuitive explanation?
Thank you.
- 387
26
votes
6
answers
14k
views
Deriving inverse of Hilbert matrix
The Hilbert matrix is the square matrix given by
$$H_{ij}=\frac{1}{i+j-1}$$
Wikipedia states that its inverse is given by
$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...
- 1,254
26
votes
5
answers
1k
views
Condition for a matrix to be a perfect power of an integer matrix
I have a question that seems to be rather simple but for I got no clue so far.
Let's say I have a matrix $A$ of size $2\times 2$ and integer entries. I want to know if there is a kind of test or ...
- 1,889
25
votes
16
answers
4k
views
functions satisfying "one-one iff onto"
Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...
23
votes
1
answer
1k
views
Is it consistent with ZF that $V\to V^{\ast \ast}$ is always surjective?
In a comment to a recent question, Jeremy Rickard asked whether it is consistent with ZF that the map $V \to V^{**}$ from a vector space to its double dual is always surjective. We know that "...
- 82.7k
22
votes
4
answers
5k
views
Eigenvalues of permutations of a real matrix: can they all be real?
For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
- 13.4k
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 ...
- 23.3k
21
votes
4
answers
4k
views
Rings over which every module is free
We know that modules over skewfields are free. Is the converse true? In other words, is it true that a nontrivial ring over which every module is free is a skewfield?
If the ring A is commutative, ...
- 1,069
21
votes
4
answers
9k
views
Condition for two matrices to share at least one eigenvector?
Suppose that I have two matrices $A$ and $B$, and I want them to share a common eigenvector $x$. For simplicity let's just assume that the eigenvalue associated with $x$ is $1$ for both matrices, so $...
- 403
21
votes
9
answers
19k
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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
20
votes
4
answers
2k
views
Nuances Regarding Naturality
It's frequently said, informally, that a natural isomorphism is one that doesn't depend on arbitrary choices.
But the phrase "arbitrary choices" lends itself to different interpretations. Consider ...
- 23k
20
votes
4
answers
2k
views
Does Anyone Know Anything about the Determinant and/or Inverse of this Matrix?
The matrix I am inquiring about here is the $n \times n$ matrix where the entry $A_{ij}$ is $\frac{1}{(i+j-1)^2}$. The $2 \times 2$ matrix looks like
$$
\begin{pmatrix}
1 & 1/4 \\
1/4 & 1/9
\...
- 209
19
votes
2
answers
1k
views
Exponentiation of vector spaces?
This question occurred to me while thinking on another one here, Name for an operation on matrices?
Can one define in an invariant way a binary operation on finite-dimensional vector spaces - let us ...
- 18.4k
19
votes
2
answers
9k
views
Distributing the Hodge map over the wedge product
Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to "...
- 191
19
votes
1
answer
2k
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Smallest eigenvalue of a tricky random matrix
While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
- 28.6k
19
votes
1
answer
895
views
Are the only local minima of $\angle(v, Av)$ the eigenvectors?
Let $A$ be an invertible $n \times n$ complex matrix. For $v \in \mathbb{CP}^{n-1}$, define
$$d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle \...
- 156k
19
votes
1
answer
856
views
A possible extension of a determinant inequality
It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$
I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
- 1,748
19
votes
1
answer
3k
views
Non-degenerate alternating bilinear form on a finite abelian group
I asked this question on math.stackexchange yesterday, but nobody has helped so far, and only 44 people have seen it! So I hope people do not mind me asking it here...
Let $A$ be a finite abelian ...
- 831
18
votes
1
answer
1k
views
How fast can extreme eigenvalues of the average of random matrices converge to their expectation?
Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
- 181
18
votes
2
answers
5k
views
Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
- 345
17
votes
6
answers
3k
views
Does the linear automorphism group determine the vector space?
I was recently thinking about what it means to put structure on a set. It seems to me that, in my area (representation theory), the two main ways of imposing structure on a set $X$ are:
...
- 12.9k
17
votes
3
answers
3k
views
Linear algebra from the categorical point of view
Is there any book of Linear algebra in the modern language of Category theory?
I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...
- 545