35
votes
Accepted
Consequences of eigenvector-eigenvalue formula found by studying neutrinos
The OP asks about generalisations and applications of the formula in arXiv:1908.03795.
$\bullet$ Concerning generalisations: I have found an older paper, from 1993, where it seems that the same ...
- 149k
33
votes
Accepted
The sum of squared logarithms conjecture
Is there anything wrong with the following argument?
First of all, by scaling all $x_i$ and all $y_i$ by a positive constant, we may safely assume that $\prod_i x_i = \prod_i y_i =1$.
The result ...
- 5,096
27
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
For $n>2$, the span of the matrices in $\mathrm{SO}(n)$ is the full space $M_n(\mathbb{R})$ of $n$-by-$n$ matrices with real entries.
One proof is using representation theory: If we let $S\subset ...
- 98.1k
22
votes
Expected value of determinant of simple infinite random matrix
Very nice problem!
Let me recall you that the determinant of $n \times n$ matrices with entries in $\{0,1\}$ is related to the one of $n+1 \times n+1$ matrices with entries in $\{-1,+1\}$: replace ...
- 1,765
20
votes
What is the time complexity of truncated SVD?
@ user40484 , fortunately your estimate for the complexity of SVD is not optimal. Otherwise, you put unemployed specialists in image compression. The complexity is in $O(\min(mn^2,m^2n))$.
Assume ...
- 2,920
19
votes
Is the determinant the only multiplicative matrix function?
It depends on what is the target space. Linear representations of ${\bf Gl}_n(k)$ do satisfy $\rho(AB)=\rho(A)\rho(B)$ by definition, and they often extend in a natural way to ${\bf M}_n(k)$.
On ...
- 47.5k
18
votes
When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?
Yes. The norm of a positive definite matrix does not exceed its trace, and the sum of traces is finite, since the sum of diagonal elements is finite for each of $n$ places.
- 88.5k
18
votes
Differentiability of Eigenvalues - Perturbation Theory
The complete reference is Kato's book Perturbation theory .... But perhaps you need only the most basic results. Then see my book Matrices (Springer GTM #216), 2nd edition. This is Section 5.2.
Mind ...
- 47.5k
17
votes
The sum of squared logarithms conjecture
Lev's proof reminded me of two papers, and unless I'm doing something silly, the said conjecture follows as a corollary of those papers. The answer below is just meant to supplement Lev's result, and ...
- 27.8k
17
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
Let $S$ be the span of $SO(N)$ .
Then it's obvious that if $A \in S$ and $D_1, D_2 \in SO(N)$ then $D_1^{-1} A D_2 \in S$ .
Therefore it's enough to show that $B := diag(1,0,...0) \in S$ .
If $N$ ...
- 2,653
17
votes
Are unitarily equivalent permutation matrices permutation similar?
Here's a direct proof that if permutation matrices $A,B$ (of size $n\ge 0$) have the same characteristic polynomial (or equivalently are linearly equivalent, since these are diagonalizable) then they ...
- 51.6k
16
votes
Accepted
On the positivity of matrices
Such matrices are called copositive in the literature.
Moreover, the statement you want to show is known to be false.
While I don't recall a counterexample right away,
an intuition for this is that ...
- 12.9k
16
votes
Are unitarily equivalent permutation matrices permutation similar?
The answer is yes by an old theorem of Brauer. In fact, permutation matrices are conjugate over any field if and only if they are conjugate by permutation matrices. See this beautiful proof by ...
- 33.7k
15
votes
Parametrization of positive semidefinite matrices
To get a parameterization of the kind you want, the space $S_{n,r}$ of positive semidefinite symmetric $n$-by-$n$ matrices of rank $r$ (with ($0<r<n$) would have to be contractible, but it is ...
- 98.1k
15
votes
Accepted
is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?
Yes, Brendan McKay showed that almost all trees have mates that are simultaneously cospectral in both adjacency and Laplacian spectra. And more.
http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees....
- 10.8k
15
votes
Accepted
When does the determinant distribute over addition?
let me assume $A$ is invertible, then you ask when
$$\det(1+X)=1+\det X,\;\;X=A^{-1}B $$
so if $X$ has eigenvalues $x_i$, $i=1,2,\ldots n$, you would need
$$\prod_{i}(1+x_i)=1+\prod_i x_i$$
basically ...
- 149k
15
votes
Accepted
Closed form solution for $XAX^{T}=B$
$B^{-1/2}XAX^TB^{-1/2}=I$, so $B^{-1/2}XA^{1/2}=Q$ must be orthogonal. On the other hand, for any orthogonal $Q$, it is simple to verify that $X = B^{1/2}QA^{-1/2}$ solves the equation, so this is a ...
- 17.7k
14
votes
Accepted
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
Elementary proof. The linear space $E$ spanned by $SO_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO_n$. Let $M=SR$ be a polar ...
- 47.5k
14
votes
Accepted
Rank of $A\otimes B - B\otimes A$
For generic $A,B$ the matrix $A$ is invertible and $B=CA$, where $C$ is also generic. We have $(A\otimes B-B\otimes A)(u\otimes v)=Au\otimes CAv-CAu\otimes Av$. The vectors $Au$ run over the whole $\...
- 88.5k
13
votes
Consequences of eigenvector-eigenvalue formula found by studying neutrinos
I want to add to the list of places where this identity has been used previously.
For a recent one, see arXiv:1710.02181 “State transfer in strongly regular graphs with an edge perturbation”.
...
- 11.8k
12
votes
Accepted
Finding the nearest matrix with real eigenvalues
I have no idea what is going on, but your conjecture is not correct. This is more transparent perhaps in the complex version. Consider
$$
A = \begin{pmatrix} i & a \\ 0&-i \end{pmatrix} , \...
- 16.7k
12
votes
Accepted
Is every real matrix conjugate to a semi antisymmetric matrix?
Yes. Every matrix can be written as the sum of a symmetric plus an antisymmetric one: $A = \frac{A+A^T}{2}+\frac{A-A^T}{2}$. Now change basis such that the symmetric part is diagonal.
- 17.7k
12
votes
Matrix elements of exponential of tridiagonal matrices
Yes! Most methods to compute exponentials of large sparse matrices are based on computing directly $\exp(A)b$ for a given vector $b$ rather than the full matrix $\exp(A)$. Just take $b$ as a vector of ...
- 17.7k
12
votes
Differentiability of operator norm
It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| \leq 1$ and $|t|$ for $|t| > 1$.
However, $\|A + ...
- 37.4k
11
votes
Accepted
No arbitrary product of matrices has eigenvalue 1?
In the case $n=4$ you could have $D = \pmatrix{0 & 0 & 0 & 1\cr 0 & 0 & 1 & 0\cr
0 & 1 & 0 & 0\cr 1 & 0 & 0 & 0}$, in which case $A_1 A_2 A_3 A_4 A_1 ...
- 51.6k
11
votes
Accepted
When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?
You can bound $\|A_k\| \leq C(n)\max_{i,j} |(A_k)_{ij}|$ for some function of the dimension only $C(n)$, because all norms are equivalent in finite dimension. If I am not mistaken $C(n)=\sqrt{n}$, but ...
- 17.7k
11
votes
Accepted
Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?
We can do this by a calculation. The assumptions on the determinant and trace are equivalent to having eigenvalues $\lambda,1/\lambda$, with $\lambda>1$. We can rotate the first eigenvector to the $...
- 16.7k
10
votes
Accepted
Product of a Finite Number of Matrices Related to Roots of Unity
The following is a conjectured generalization of the claimed identity which may help in proving it. We prove this generalization (and hence also the identity from the question) in the case that $3$ ...
- 16k
10
votes
Accepted
Decomposing a matrix into a product of sparse matrices
Given an invertible $n \times n$ matrix $\mathrm A$, we perform Gaussian elimination until we obtain a (nonsingular) diagonal matrix. In other words, we left-multiply $\mathrm A$ by permutation ...
- 2,162
10
votes
Accepted
Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for scalar $p$
Write $Y=SDS^{-1}$, where $D$ is diagonal (since $X_1$ and $X_2$ are psd, $Y$ is diagonalizable). Then, observe that
\begin{equation*}
f(p) = \text{tr}(S^{-1}(I+pSDS^{-1})^{-1}SS^{-1}(I-p^2SDS^{-1})...
- 27.8k
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