All Questions
6,291 questions
8
votes
4
answers
1k
views
Subspaces of End(V) that can fix any vector
Suppose V is a finite-dimensional vector space and I have a linear subspace of its endomorphisms
$$W \subseteq \mbox{End}(V).$$
How can I easily check if every vector of $V$ is fixed by some element ...
- 5,898
2
votes
1
answer
300
views
Rayleight Ritz Ratio and smallest eigenvalue for a set of given matrices
I am familiar with Rayleigh Ritz Ratio for hermitian matrices. Let $A_1$ be a given $N \times N$ hermitian matrix. Then the smallest eigenvalue of $A_1$ is given by
\begin{align}
\lambda_{min}(A_1)=\...
- 1,421
1
vote
0
answers
880
views
Order of column vectors in jordan normal form
When constructing the jordan normal form
of the matrix $A$ one has to construct the jordan blocks corresponding to the eigenvalues $\lambda_i$ of the matrix.
If the eigenvalue is simple or semi-simple ...
- 21
10
votes
1
answer
1k
views
Relationship between eigenvalues of $A-B$ and eigenvalues of $A^2-B^2$
Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying
$$c \leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$$
and
$$c \leq \lambda_{\min}(B_n)\leq \...
- 103
5
votes
1
answer
2k
views
Condition for block symmetric real matrix eigenvalues to be real
I have a $2n \times 2n$ block symmetric matrix that in the simplest case ($n=2$) looks like:
$$
M_2 = \begin{bmatrix}
a_1 & 0 & b_{1,2} & -b_{1,2}\\\
0 & -a_1 & b_{1,2} & -b_{...
- 151
11
votes
1
answer
2k
views
Sum of commuting semisimple operators
Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a $T$-invariant complement(for algebraically ...
- 2,233
4
votes
1
answer
411
views
power log distance between matrices
In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of ...
- 96.4k
2
votes
1
answer
486
views
Is there any connection between this matrices
Matrices I discuss are all $N\times N$ hermitian matrices. Define two positive (semi)definite matrices $H_1$ and $H_2$. Define the following matrices
\begin{align}
P_1&=H_1+(I+H_2)^{-1} \\\
P_2&...
- 1,421
9
votes
2
answers
366
views
Iterating Random Matrix Operations
Consider the following probability measure on the integers concentrated around $0$: the probability of drawing $0$ is $\frac{1}{2}$, of drawing ($1$ or $-1$) is $\frac{1}{4}$ split evenly among the ...
- 15.5k
15
votes
1
answer
2k
views
Necessary and sufficient conditions for a sum of idempotents to be idempotent
Given: a finite list of $n$-by-$n$ idempotent complex matrices $E_1, E_2, \ldots, E_k$.
If all pairwise products $E_i E_j$ (with $i \neq j$) are zero, it is trivial to show the sum $E_1 + E_2 + \cdots ...
- 151
9
votes
1
answer
1k
views
Determinantal formula for the nullspace of a singular matrix
In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that I feel I should ...
- 82.7k
8
votes
1
answer
1k
views
Norm of inverse confluent Vandermonde matrix
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as
$$V=
\begin{bmatrix}
v_{1,0}&v_{2,0}&\dots&...
- 959
2
votes
0
answers
184
views
Checking for error in conjugate gradient algorithm
What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
- 489
0
votes
1
answer
147
views
Name for a particular subgroup of parabolic subgroups of the general linear groups. [duplicate]
Possible Duplicate:
Name for a particular subgroup of parabolic subgroups of the general linear groups.
Let $V$ be vector space. The subgroup $P$ of $GL(V)$ consisting of all automorphisms ...
- 19
5
votes
2
answers
708
views
Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers
Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...
- 32.5k
8
votes
3
answers
1k
views
Relating a Polynomial equation to the characteristic equation of a Hermitian matrix
This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
- 1,421
1
vote
0
answers
263
views
Average weighted value of a linear functional over increasing bounded subsets of Z^n
Say you're working within the finite-dimensional free Z-module $\mathbb{Z}^n$, and you want to impose a "norm" on this module. By a "norm" I mean a function $\|·\|: \mathbb{Z}^n \to \mathbb{R}$ which ...
- 4,907
0
votes
1
answer
272
views
A certain type of quadratic problem.
I am interested in solving the following equality constrained quadratic (?) problem.
\begin{align}
\min_{u^{H}u=1}~(u^{H}A_1u) \\\
s.t.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ ...
- 1,421
21
votes
0
answers
904
views
Cauchy matrices with elementary symmetric polynomials
$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...
- 28.6k
4
votes
2
answers
604
views
A certain type of quadratic constrained quadratic program (QCQP)
Let $P_1$, $P_2$ be two Hermitian matrices. Can anyone comment on the following QCQP?
$$\begin{array}{ll} \text{minimize} & z^{H} z\\ \text{subject to} & z^{H} P_1 z +1 \leq 0\\ & z^{H} ...
- 1,421
0
votes
0
answers
237
views
Geometric Mean of Positive Matrices
Hello all,
My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices $(A,B)$ is given by:
$M_0(A,B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-...
- 155
4
votes
0
answers
167
views
Reduction of size of orthogonal matrices.
While experimenting with orthogonal vectors I've noticed the
following transformation: If
$$
A = \begin{bmatrix}z & r \cr
c & B\end{bmatrix}
$$
is orthogonal, $z$ ...
- 798
3
votes
1
answer
294
views
Decomposability of exterior two-forms
Hello,
The following question appears as a step in my proof. It seems easy but somehow I have not been able to prove this. I could solve few special cases though. Any help in this context is welcome....
- 895
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
$$d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle \...
- 156k
3
votes
3
answers
190
views
Simultaneous "Monomialization" of a set of operators.
We all know that a set of commuting diagonalizable matrices can be simultaneously put in diagonal form. My general question is:
Under what conditions can a set of (diagonalizable) matrices be ...
- 1,639
1
vote
1
answer
177
views
if $\Pi_1$ and $\Pi_2$ be elliptic planes then $\Pi_1 \oplus \Pi_2 $ is still elliptic?
Let $\Omega \in\Lambda^{4}\big(V^{ \star }\big)$ be volume form. Define symplectic bilinear form
$q: \Pi \oplus \Pi \rightarrow R $
$\big( \alpha ,\beta \big) \longrightarrow \alpha \...
user21574
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
5
votes
4
answers
5k
views
Eigenvectors of the Fourier matrix
Let $n$ be a positive integer.
The $n$ by $n$ Fourier matrix may be defined as follows:
$$
F^{*} = (1/\sqrt{n}) (w^{(i-1)(j-1)})
$$
where
$$
w = e^{2 i \pi /n}
$$
is the complex $n$-th root of ...
- 1,641
2
votes
1
answer
210
views
Gram matrix modulo 4
Suppose we have a full rank, integer sublattice $L$ of the integer lattice $\mathbb Z^d$, where we fix the dimension $d$. Consider the Gram matrix $M$ of $L$, relative to some basis for $L$, and ...
- 623
1
vote
0
answers
243
views
Norm bound of the entrywise logarithm of a stochastic matrix stationary matrix
Hello,
Denote $\log_\star$ as the entrywise logarithm operation, and let $A$ be some row-stochastic matrix such that $\lim_{p\rightarrow\infty}A^p$ exists and all its entries are non-zero.
As a part ...
- 225
0
votes
0
answers
224
views
When does a real-valued function of a matrix depend only on eigenvalues?
Let $\mathcal{N}$ be the space of all $n \times n$ matrices that are similar to some nonnegative matrix with zero diagonal and let $f: \mathcal{N} \to \mathbb{R}$ be a continuously differentiable ...
- 1,068
2
votes
1
answer
2k
views
power of a block triangular matrix
I have a matrix in the form :
$$M =
\begin{pmatrix}
A & 0 & 0 \\\
B & A & 0 \\\
C & D & A
\end{pmatrix}
$$
where $A,B,C,D$ are diagonalizable square matrices and I want to ...
- 21
0
votes
1
answer
1k
views
Function (/matrix) to generate linearly independent vectors.
Hi,
I want to whether there is a vector generating function (/matrix) such that it can generate a m-dimensional vector which will always be linearly independent of the set of m-dimensional vectors ...
3
votes
1
answer
258
views
Subharmonic envelope
I came across a more complicated version of the following problem. It is so elementary, I think that there had to be some research done on this in the past. If someone has any ideas please let me know....
- 1,211
1
vote
1
answer
309
views
Scaling laws for singular values of random matrices
Assume that we have an $n\times n$ matrix ${\bf A}$ with elements drawn i.i.d. Gaussian with mean zero and variance 1.
Are there any results on the asymptotic behavior of its $i$-th largest singular ...
- 449
3
votes
0
answers
250
views
multidimensional rotation terminology
Given an element $g$ of the orthogonal group $O(n)$, is there a name for the subspace of $R^n$ that's fixed by $g$, and a name for the orthogonal complement of this space? (The latter is what I ...
- 19.7k
8
votes
1
answer
7k
views
Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive
Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
- 83
20
votes
3
answers
2k
views
Approximating commuting matrices by commuting diagonalizable matrices
Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that
$A_n \rightarrow A$, $B_n \rightarrow B$.
Each $A_n$ is diagonalizable and the same for ...
- 571
12
votes
2
answers
1k
views
Quadratic Farkas' Lemma?
The Farkas Lemma says that if a system of linear inequalities implies
yet another linear inequality, then this last inequality can be obtained by
taking a positive linear combination of the ...
- 23k
2
votes
1
answer
186
views
Can I bound the degree of a contracting homotopy in an exact filtered complex?
Suppose that I have given you a bigraded vector space $V = \bigoplus_{i,j} V_{i,j}$. The first grading is a "homological" $\mathbb Z$-grading, and the second is an independent $\mathbb Z$-grading. ...
- 54.6k
5
votes
2
answers
429
views
Simultaneous maximization of two Generalized Rayleigh Ritz Ratios
Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
- 1,421
3
votes
2
answers
501
views
Lattice reduction on an orthonormal lattice?
Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you (...
- 8,688
2
votes
1
answer
1k
views
Trace inequality for matrices with determinant 1
Let $A$ and $B$ be two matrices with $\det(A)=\det(B)=1$. Does it follow that
$\sqrt{\mathrm{tr}(A^TB^TBA-I)}\le\sqrt{\mathrm{tr}(A^TA-I)}+\sqrt{\mathrm{tr}(B^TB-I)}$
I suspect that this can be ...
- 320
0
votes
1
answer
1k
views
Simultaneous Jordanization
Hello everyone
I would like to have a detailed reference to the statement bellow:
Let $A,B\in \mathbb{R}^{n\times n}$ such that $AB=BA$. Suppose $A$ has real eigenvalues only and $B$ is ...
3
votes
2
answers
546
views
Matrix subspace with full rank
Suppose two integers $n< m$, and one is given a matrix subspace of $n\times m$ complex matrices, says $S$.
I am asking for algorithms or conditions which can answer the following question:
...
- 1,503
6
votes
1
answer
276
views
Linear algebra over principal rings
Consider an extension $R\subseteq S$ of commutative rings, and suppose that $R$ is principal (i.e., $0$ is the only zero-divisor of $R$ and every ideal of $R$ has a generating set of cardinality $1$). ...
- 6,700
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
0
votes
1
answer
165
views
Nonlinear matrix equation 2
Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$
$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \...
- 41
0
votes
2
answers
905
views
fixed points of system of quadratic equations
Let $\Phi: R^n \to R^n$ satisfy
$\Phi(x)=u+Ax+Q(x)$, with $x=(x_1, x_2,\ldots, x_n) \in R^n$. $u$ is a given positive vector, $A$ non negative matrix, and $Q(x)$ quadratic mapping with
$Q(x)_i=x_i(...
- 1
4
votes
2
answers
2k
views
Probability of a submatrix to be full rank in a N x N Random Matrix of rank m.
Consider a random matrix $\mathbf{A} \in \mathbb{C}^{N \times N}$ of rank $m$ with $m < N$ that follows the Wishart distribution ( http://en.wikipedia.org/wiki/Wishart_distribution ).
I have a ...
- 43