All Questions
Tagged with linear-algebra operator-theory
74 questions
14
votes
5
answers
2k
views
How far is a set of vectors from being orthogonal?
Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones?
Or, more formally...
Suppose $...
- 1,513
12
votes
6
answers
7k
views
Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?
I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but ...
- 663
12
votes
1
answer
1k
views
Decomposition of positive definite matrices.
It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...
11
votes
2
answers
2k
views
Operator that commutes with projections
We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$
Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
- 536
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
11
votes
2
answers
353
views
Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495
10
votes
2
answers
2k
views
How to transform matrix to this form by unitary transformation?
Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e.
$$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$
Then is ...
- 273
9
votes
2
answers
912
views
A Matrix Inequality for positive definite matrices
Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|\leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^{\frac{...
- 1,879
9
votes
1
answer
535
views
Well known matrix inequality?
I suspect that the following matrix inequality is well known, but I can't find a reference or proof:
Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true?
$${...
- 1,211
8
votes
3
answers
691
views
Commutant of the conjugations by unitary matrices
Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
- 12.6k
8
votes
0
answers
333
views
Monotonicity of log determinant of Gaussian kernel matrix
Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation}
be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...
- 55
7
votes
1
answer
635
views
Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$
This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "...
- 715
6
votes
1
answer
1k
views
Inequality between nuclear norm and operator norm for positive definite matrices
I will use $\|\|_*$ to denote the nuclear norm (sum of singular values) and $\|\|_2$ to denote the operator norm / matrix 2-norm (largest singular value).
Consider two positive definite $n \times n$ ...
- 61
5
votes
2
answers
276
views
Dilation of bounded linear operators
Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
- 51
5
votes
4
answers
839
views
Norm bounds on spectral variation and eigenvalue variation
Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively.
The spectral variation of $B$ w.r.t. $A$ and the eigenvalue variation of $B$ and $A$ are, respectively,
\begin{...
- 43.2k
5
votes
0
answers
330
views
Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 153
5
votes
0
answers
251
views
Eigenvalues of a certain product of matrices with special structure
Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = [G\hspace{1em}\...
- 6,853
4
votes
2
answers
159
views
$\sum_{k=1}^dA_k^*A_k$ and $\sum_{k=1}^dA_kA_k^*$ have the same norms if $A_k$ are commuting
Let $E$ be a complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all operators on $E$.
Let $A_1,\cdots,A_d$ be pairwise commuting operators on $E$. Is the equality
$$\left\|\displaystyle\...
- 1,154
4
votes
1
answer
670
views
Is the product of projection matrices diagonalizable?
Let $P_i=V_{i}V_{i}^{\top}\in\mathbb{R}^{m\times m}$ where $\forall i\in[T]: V_{i}\in\mathbb{R}^{m\times n}$ is a “tall” matrix (i.e., $m \ge n$) with orthonormal columns. Note that these matrices are ...
- 673
4
votes
1
answer
875
views
comparing norms of tensor product of two Hilbert spaces
Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$
consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ...
- 41
4
votes
1
answer
54
views
Krein-Rutman for integral transforms: proof of convergence to leading eigenvector
Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory.
Consider an integral ...
- 312
4
votes
1
answer
353
views
When is rank-1 perturbation to a positive operator still positive?
Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
- 695
4
votes
1
answer
591
views
Derivative of trace
Consider two positive-semi definite matrices $T_1, T_2$ of unit trace.
Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two.
We then study $f(\lambda) := \operatorname{tr}(T(\...
- 536
4
votes
0
answers
311
views
Estimates of the Frobenius norm of commutator
Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
- 949
4
votes
0
answers
96
views
On decidability of an infinite dimensional dynamic system
Consider two infinite dimensional Toeplitz integer matrices $A,B$ where $B$ is just a shift operator and an infinite dimensional vector $V_0$.
Given a prime $p$ and an infinite dimensional integer ...
- 13.9k
3
votes
2
answers
113
views
Behavior of orbits under small perturbations
Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange.
Suppose $T$ is a bounded linear operator on $l_2$ and $x\in l_2$ is a ...
- 1,361
3
votes
2
answers
367
views
norm of the matrix series
The goal is to obtain an upper bound for the norm of the vector
$$
\left\|\sum\limits_{k=0}^{\infty}(I−A)^kAw_k\right\|
$$
for any symmetric matrix $A\in{\mathbb R}^{n×n}$ which $0\preceq A\preceq I$ ...
- 31
3
votes
1
answer
415
views
Inverse of block matrix
Let $V$ be a finite-dimensional vector space and consider the space $X=V\times V\times V\times V.$
Consider the block matrix
$$A = \begin{pmatrix} A_1 & A_2 \\ A_2^* & -A_1\end{pmatrix}$$
...
- 192
3
votes
1
answer
741
views
Operator norm of difference of matrix decompositions
This question is in part related to a question that I have already posed.
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
- 55
3
votes
1
answer
83
views
Detecting linear operator from actions of powers on subspace
Say I have a sequence of linear operators $A_1,...,A_n$ on a (real) vector space $V_1$. I suspect that there's a second vector space $V_2$, and an operator $A$ on $V_1\oplus V_2$, such that $A_i=(\...
- 4,593
3
votes
1
answer
119
views
Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed
The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n \...
3
votes
0
answers
198
views
On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
3
votes
0
answers
77
views
Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
3
votes
0
answers
234
views
Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $
What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
- 923
3
votes
0
answers
119
views
Increasing sequence of closed subspaces of $L^2$ and error estimate of a product of orthogonal projections
We define an increasing sequence of closed subspaces
\begin{align*}
V_{0} \subset V_{1} \subset V_{\ell} \subset \dots
\end{align*}
of $L^2(I)$ where $I=(0,x_{max})$, and each $V_{\ell}$ is equipped ...
- 131
3
votes
0
answers
261
views
Generalization of trace and associated determinant
The standard relation between the trace and the determinant of matrices is presented in the MO-Q "Cycling through the zeta garden" where the log and exp functions allow one to jump between additive ...
- 10.5k
2
votes
1
answer
215
views
Forming real positive semidefinite matrices from complex matrices
I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices.
Let $Q \in \...
- 31
2
votes
1
answer
358
views
The operator equation $AB = \lambda BA$ for self-adjoint operators
Suppose that $A$ and $B$ are self-adjoint bounded linear operators on a Hilbert space and $\lambda \in \mathbb{C}$. It turns out that if $\lambda \notin \{-1, 1\}$ then $AB=\lambda BA \implies AB = ...
- 3,699
2
votes
2
answers
170
views
Given a subdomain of GL(n), when is the map from matrices to their matrices of eigenvectors a diffeomorphism?
I'm wondering if there are any general conditions on a subdomain of $GL(n)$, which would guarantee that the map from a matrix to its matrix of eigenvectors is a diffeomorphism.
For example, given a ...
- 173
2
votes
1
answer
1k
views
Norm of a the skew symmetric part of a unitary matrix
Let $U \in \mathbb{R}^{n \times n}$ be a unitary matrix, $U$ can be nonsymmetric, its eigenvalues can be complex numbers and all have modulus $1$.
Is there an upper bound for the maximum singular ...
- 323
2
votes
1
answer
141
views
On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
- 4,058
2
votes
1
answer
126
views
Determinant of a matrix involving the Prolate Spheroidal Wave Functions
The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation:
$$\int_{-T}^T\varphi_n(x) \text{sinc}(t-x) dx = \lambda_n \varphi_n(t)$$
where $\text{sinc}(t) = \sin(\pi ...
- 93
2
votes
0
answers
60
views
Perron-Frobenius theory for operators on matrices
Let $A$ be a Hermitian linear operator on the space of $n\times n$ complex matrices. Let's suppose $A$ is "non-negative" in the sense that it preserves the cone of non-negative definite (...
- 53
2
votes
0
answers
81
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,058
2
votes
0
answers
267
views
Example of a unital contractive map that is not completely positive on an operator system
I am aware of maps that are positive but not completely positive (for example transpose map). BUT I can not think of an example of the following type.
Does there exist an operator $T$ such that a map $...
- 231
2
votes
0
answers
506
views
Finding a basis for the range of a linear function
I realize this question is not high level but I have posted it on Math Stackexchange:
Stackexchange question
and have received some upvotes but no answers or comments, so I am trying here.
I will need ...
- 121
2
votes
0
answers
257
views
The nonlinear operator defined as the commutator of a matrix and a nonlinear operator
In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up:
Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...
- 620
2
votes
0
answers
487
views
What is the trace of this operator in $L^\infty$ (if this question make sense)?
Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator :
\begin{array}{...
- 283
1
vote
1
answer
220
views
Dimension of commutant
Suppose that $A = M_n(\mathbb{C})$ be the algebra of $n*n$ matrices over $\mathbb{C}$.
If com(A) = {$B \in M_n(\mathbb{C}); AB = BA$}, then what is the $dim(com(A))?$
- 581
1
vote
1
answer
1k
views
Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...
