Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,874 questions
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Bounds on spectral norm of difference of powers of linear operators
Given bounded linear operators $A$ and $B$, with $||A-B||<\epsilon$, what is the tightest upper bound on $||A^n-B^n||$, for positive integer $n$? Additionally, for my specific situation, $A$ is ...
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Nonnegative locus of the class of positive semidefinite symmetric bilinear forms
Let $\mathcal{P}$ denote the class of positive semidefinite symmetric bilinear forms $A : \mathbb{R}^{n} \times \mathbb{R}^n \to \mathbb{R}$. Let
$$
X(\mathcal{P}) := \{(x,y) \in \mathbb{R}^n \times \...
user94803
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Do these matrices have the same null space?
Let $[\theta_1,\theta_2, \dots, \theta_N]^\mathrm{T} \, \in \mathbb{R}^N$. The angles are not all identical (on the circle), i.e. $[\theta_1,\theta_2, \dots, \theta_N] \not \equiv c [1,1,\dots, 1]^\...
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Matrix series as block matrix times vector of matrix multiplication
Let
$V_i \subset \mathbb{R}^n$ and $V_0 \supset V_1 \supset ... \supset V_i \supset ...$,
$A_i, B_i: V_i \rightarrow V_i$ be square non-symmetric positive definite matrices,
$Q_i:V_{i-1}\...
- 323
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Perturbation theory for matrices
I encountered the following problem. Since this is somewhat not related to what I normally do, I wanted to know what the best estimates in this field are.
Let $A \in \mathbb{R}^{n \times n}$ be a ...
- 3
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198
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The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
- 199
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How to calculate the inverse of the sum of kronecker products with the identity matrix
How to calculate $G^{-1}$ efficiently when $G$ is a large matrix knowing that:
\begin{eqnarray}
G=I⊗A + A⊗I
\end{eqnarray}
Or since i'm using $G^{-1}$ to multiply by some other matrix, how to find $...
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142
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sub-space restricted minimum eigenvalue of Hadamard product of two PSD matrices
Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{n\times n}$ be two positive semidefinite matrices. Also let $\mathbf{A}\circ \mathbf{B}$ denote the Hadamard product of $\mathbf{A}$ and $\mathbf{B}$. A ...
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Recurrence Equation and Matrix Convergence
To begin with, let us give the conceptual background needed to expose the problem. First of all, we shall consider the set $\mathbb{L}^{n} = \mathbb{R}^{n}_{\geq0} = \{\overrightarrow{x}\in\mathbb{R}^{...
user89024
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Non-strict column diagonally dominant matrix inner product
Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is:
$$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$
where $0 \le a_{j,j} \le 1$ and $-1 \le ...
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SVD alternatives for symmetric matrices
Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices
$$
A = E'E
$$
Practically this can be done easily using SVD ...
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174
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The effect of linear transformation on generic vectors [closed]
I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$.
I know that if $M$ has p-norm $\|M\|_p = \lambda$, then by ...
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a sum of ratios of quadratic forms
I have the following function that I would like to optimize over the value A
$$f(A)=\sum_k \frac{\mathbf{y}_k^H\left[\begin{array}{cc} 1&0\\ 0& A \end{array} \right]\mathbf{x}_k\mathbf{x}_k^H\...
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Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?
Let $\varphi_k\in\mathbb{C}$ be a primitive $k$-th root of unity, and define the sets
$$S_\ell:=\left\{\left[\begin{matrix}x\\x\varphi_k^\ell\end{matrix}\right]\in\mathbb{C}^{2n}\;\middle|\;x\in\...
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Simultaneous special orthogonal similarity problem
Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
$$A=...
user76479
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2
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588
views
Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation
Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
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371
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Is there relation between vector valued RKHS and interpolation space?
Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: https://en....
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The condition number of a scaled Vandermonde matrix
Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and
let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$.
My intuition says that the condition number should be invariant under ...
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Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]
Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$?
Cross posted from ...
- 2,099
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Symmetric matrix from a nonsymmetricc matrix
Basically this is a part of a long algorithm to calculate some matrix properties.
Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...
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1
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124
views
Stationary distribution of random walk alias solving uncountably many linear equations [closed]
Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$.
Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...
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72
views
Characterisation of a matrix ordering property
Let $n$ be a positive integer; we consider all matrices mentioned henceforth to be $n$-by-$n$ matrices. Let $A$ and $B$ be matrices wherein all entries are nonnegative (such matrices will be called ...
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118
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Action of $(\mathbb{Z}/2g\mathbb{Z})$ on quadratic forms on $\mathbb{Z}/2\mathbb{Z}$-vector space
Let $\mathbb{Z}/2\mathbb{Z}$ the 2 elements field, with additive notation.
I need some clarifications on the relation between quadratic forms on a $\mathbb{Z}/2\mathbb{Z}$-vector space (say, of ...
- 3,779
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456
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Integral over Kronecker product
Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that
$$ \int_{0}^T A(t) dt $$ is invertible.
Does it then follow that the matrix
$$ \int_{0}^T A(t) \otimes A(...
user45183
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How do eigenvalues change if we duplicate a row and column of a symmetric matrix?
Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix,
$$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
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546
views
Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
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664
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Kneser graphs eigenvalues
Basically, I want to prove that, in the Kneser graph (wikipedia has a good definition),$K_{n, m}$, if $n_{-}(A(G)) $ and $n_{+}(A(G))$ denote the number of negative and positive eigenvalues of A(G) ...
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203
views
Eigenvalues of a given parametrized matrix.
Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as
\begin{align}
\mathbf{C}=\left(\mathbf{I}*\frac{1}{\...
- 1,421
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1
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720
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Perturbation of Cholesky decomposition for matrix inversion
I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$
where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...
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Measuring the extent of entanglement in tensor products
Is there a non-negative integer valued grading function on the tensor product of two Hilbert spaces which measures the extent of entanglement ?
0
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1
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157
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singular values function
Let $\mathbf{F}\in\mathbb{C}^{M\times M}$ and $\mathbf{D} = \operatorname{diag}(\mathbf{d})$ where $\mathbf{d}\in\mathbb{R}^M$. By SVD, $\mathbf{F}\mathbf{D}\mathbf{F}^H=\mathbf{U}\mathbf{S}\mathbf{U}^...
- 9
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1
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2k
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eigen-decomposition solution? is it unique?
Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
- 1
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242
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does the basis in the singular value decomposition of a sum depend on the singular values of the summands
Suppose you have 4 matrices with singular value decompositions
$A = U_1 \Sigma_A V_1^{\dagger}$, $B = U_2 \Sigma_B V_2^{\dagger}$, $C = U_1 \Sigma_C V_1^{\dagger}$ and $D = U_2 \Sigma_D V_2^{\dagger}$ ...
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105
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The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)
Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries
\begin{equation}
r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; A\in\mathbb{R},l_0\in\...
- 33
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1
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How do you solve a tridiagonal matrix where all 3 diagonals are ones? [closed]
This is probably really simple, and I'm missing something, but Thomas' algorithm doesn't seem to work.
- 1
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520
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solving trace norm equality [closed]
Problem Formulation
under what conditions can we solve $\mathrm{trace}(\mathbf{AB})=0$ ? or more specifically, when will $\mathrm{trace}(\mathbf{AB})=0$ implies that $\mathrm{trace}(\mathbf{B})=0$.
...
- 125
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2
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174
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Matrices whose kernel escapes a sub-vector space
Let $n>n'\gg m$ and $V$ be a subspace of $\mathbb{C}^n$ of dimension $n'$. I am trying to characterize the set $X$ of $m\times n$ matrices $A=(a_{ij})$ satisfying $\ker(A)\not\subseteq V$, that is, ...
- 537
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170
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Tensoring with descending chain of modules
Let $A \to B$ be a ring homomorphism. Let $M_1 \supseteq M_2\supseteq \ldots$ be an infinite chain of $A$-modules ($M_i$ not necessarily finite free). Suppose that the limit $\cap_{i=1}^{\infty} M_i$ ...
- 11
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172
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orthogonality in a lattice
Let $\Lambda$ be a lattice with a quadratic form $q$ of signature (3,19).
Let $\Lambda_{\mathbb{R}}:=\Lambda\otimes \mathbb{R}$ and $W\subset \Lambda_{\mathbb{R}}$ a positive subspace of dimention 3.
...
- 107
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1
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139
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Spectrum of a Laplacianized matrix
Suppose that $A$ is a positive matrix and that we let $R$ be the diagonal matrix of $A$'s row-sums. What can be said about the spectrum of $R-A$? I am particularly interested in the largest eigenvalue ...
- 6,962
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573
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about decomposition of three forms
Patrick D. Baier in his Ph.D. thesis for proving the theorem 2.1.4 used the following non-trivial fact (in chapter 2 on page 14):
Let $0\neq X\in V$ (here $V$ is of dimension 6), $W^\ast = Ann(X)$ ...
user21574
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212
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L a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L).
Let L be a finite-dimensional complex semisimple lie algebra, then ad(L)=Der(L). (Der is short for derivation). In order to show that ad(L)=Der(L), the proof I followed proves that that the ...
0
votes
1
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156
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Projection and Positive matrices [closed]
Sorry my question was not clearly stated. I will ask it more clearly.
Let $G$ be a matrix with only nonnegative elements with linearly independent
columns. Then there exists a column, ${\bf g}$ of $G$...
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
0
votes
1
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163
views
Nonsingular zeroes are algebraic?
I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields):
Suppose $Q\in L^n$, $f_1, \...
- 1,979
0
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1
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193
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Ease of calculation of norm
I have SPD matrix A and two vectors z and b.
Is there exist a norm where I can calculate $||A^{1/2}b-z||$ without having to calculate $A^{1/2}b$ explicitly ?
- 489
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1
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148
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Augmenting $t$-dimensional sub-spaces into $t+1$-dimensional sub-spaces through a basis
Let $V_1, V_2, ..., V_n$ be $t$-dimensional sub-spaces of an $n$-dimensional vector space $V$
where $t \lt n$.
Under what conditions the following would be true:
for any $B= \{v_1, v_2, ..., v_n\...
- 1,034
0
votes
3
answers
302
views
Vector "product" diagonalization
Consider a vector space $V$ (with dimension $n+1$ and elements $v$) on which a (commutative and associative) "product" $\odot$ taking $V\odot V\rightarrow V$ is defined, and an $1$ element $v_0$ ...
- 4,829
0
votes
2
answers
156
views
inclusions of linear colimits into smooth manifolds
Let $V$ be the category of finite dimensional vector spaces and $M$ the category of
smooth finite dimensional Hausdorff manifolds.
Now suppose any finite dimensional vector space is equipped with a ...
- 2,074
0
votes
2
answers
148
views
Bounding 2nd Eigenvalue of a Pseudo-Rotation-ish matrix
Let $p,q$ be arbitrary primes.
Let $N = p * q$.
Let $I$ be the $N * N$ identity matrix.
Let $R$ be the $N * N$ matrix defined as follows:
$R[x_0 * p + y_0, x_1 * p + y_1]=1$ if and only if $x_0+1 ...
- 45