Questions tagged [matrix-decomposition]

Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.

Filter by
Sorted by
Tagged with
0
votes
1answer
19 views

How come that I approximate a matrix by a product of less order matrices and I have more elements in the end?

I'm sure I'm missing something here but it's honestly giving me a headache. First of all, by the Singular Value Decomposition theorem I can decompose any $m \times n$ matrix into: $$A = \sum_{i=1}^k\...
0
votes
1answer
33 views
+100

Is $\| B(AB)^\dagger \|_2 $ uniformly bounded for all invertible matrices $B$?

Consider $\| B(AB)^\dagger \|_2$ where $A$ is a real matrix, $B$ is a real, square and symmetric matrix, and $(AB)^\dagger$ is the Moore-Penrose pseudoinverse of $AB$. Is $\| B(AB)^\dagger \|_2$ ...
0
votes
0answers
8 views

Coordinate gradient descent for optimizing $\arg\underset{D}{\min}\{\text{tr}(DFD^T)-2\text{tr}(ED^T)\}$ [closed]

How can coordinate gradient descent be applied for solving $$\arg\underset{D}{\min}\{\text{tr}(DFD^T)-2\text{tr}(ED^T)\}$$? where $D, F, E$ are all matrixes, columns of $D$ denotes atoms of the ...
1
vote
0answers
23 views

Closeness of $ \mathbf{VV}^\top $ to $ \mathbf{I} $, $ \mathbf{V} \in \mathbb{R}^{d \times r} $ right-singular vectors of singular PSD matrix

Edit: I made a mistake in my numerical experiments, as numpy.linalg.svd returns the "canonical" full SVD, where $ \mathbf{U}, \mathbf{V} \in \mathbb{R}^{d ...
0
votes
1answer
16 views

Let $A$ be positive definite, show that there is $S\in \operatorname{GL}(n, \mathbb{R})$ such that $A=SS^T$

Let $A$ be positive definite, show that there is $S\in \operatorname{GL}(n, \mathbb{R})$ such that $A=SS^T$ I know that this hints at the Cholesky decomposition. But I don't have to prove that $A$ ...
0
votes
0answers
10 views

Numerical range of a real matrix is symmetric about the real axis

Problem If A is a real matrix (A ∈ Mn,n(R)), then the numerical range of A is symmetric about the real axis. How do I prove this? Is it something to do with Hermitian matrices?
1
vote
0answers
15 views

Find nearest postive semidefinite matrix keeping diagonal.

Given a symmetric matrix $W$, is there a way to find the nearest positive semidefinite matrix $W_{PS}$ such that it is nearest to $W$ by some criterion (e.g. Frobenius norm) but it is such that $\text{...
0
votes
1answer
22 views

Convex hull is a subset of the numerical range

Question Show that if A is a normal matrix, then W(A) (the numerical range of A) is the convex hull of the eigenvalues of A. Progress I have proved that the W(A) ⊆ conv A similar to Numerical range of ...
0
votes
0answers
7 views

Singular values of a matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows. We wish to find a tight bound on the singular values. We know on the rows that they have (...
0
votes
1answer
22 views

How do we know V in the SVD is the eigenvectors of M*M?

if we have any real matrix M nXm, the SVD (singular value decomposition) allows us to decompose it into $U{\Sigma}V^T$, where V is an orthogonal real matrix composed of the eigenvectors of $M^TM$. ...
3
votes
0answers
28 views

Matrix with different dimensions and same eigenvalues (multiplicity)

Suppose we have two positive definite symmetric matrices $A\in\mathbb{R}^{m\times m}$ and $B\in\mathbb{R}^{(m+1)\times (m+1)}$. In particular, $B\equiv \left( \begin{matrix} a_{11} & ... & a_{...
0
votes
0answers
11 views

Proper Generalized Decomposition

I have researched some topics regarding Proper Generalized Decomposition. All the text more or less state the what is the algorithms to conduct PGD as Assume that the solution is separable in x(space)...
1
vote
1answer
18 views

Decomposition of Hermitian matrix

Given a $4 \times 4$ hermitian matrix, how do I decompose the Hermitian matrix into a linear combination of unitaries?
0
votes
1answer
14 views

Augmented Matrix and Dimensions

So when we have the standard forms of a linear equation system and then we represent it as an augmented matrix, the constant is not included as part of the dimension of the augmented matrix, right? So,...
-1
votes
1answer
47 views

How to find the SVD when eigenvalue is $0$? [duplicate]

After calculating the eigenvalues, I get $1040400$ and $0$. Since one of them is $0$, how do I calculate (orthogonal) matrix $U$? $$u_i = \frac{1}{\sqrt{\lambda_i}} A v_i$$ I know there are a similar ...
0
votes
1answer
35 views

Decomposition of $\text{GL}^+(2,\mathbb{Q})$ into $\text{SL}(2,\mathbb{Z})-$left cosets

Consider the following group: $$\text{GL}^+(2,\mathbb{Q})=\{M\in\text{GL}(2,\mathbb{Q}):\det M>0\}$$ I'm trying to prove the following fact: $$\text{GL}^+(2,\mathbb{Q})=\bigsqcup_{\substack{x,z\in\...
1
vote
1answer
26 views

Does every Matrix have a REF?

I'm wondering if anyone here has experience using Wolframalpha? I can't seem to get a row echelon form (REF) or echelon form, it always does reduced row echelon form (R-REF). Anybody know how to get ...
0
votes
0answers
23 views

Proof that matrix exponent elements belong to certain span

I am to prove the following proposition using S+N decomposition. Let $A\in M_n(\mathbb{F})$ - that is, an $n\times n$ matrix over some field $\mathbb{F}$, and $\mu_1,...,\mu_l$ its eigenvalues with ...
0
votes
0answers
31 views

Solution of $(DA+AD)^T=B$ about D?

$$ \begin{aligned} \mathbf{D}_{t} & \triangleq \underset{\mathbf{D} \in \mathcal{C}}{\arg \min } \frac{1}{t} \sum_{i=1}^{t} \frac{1}{2}\left\|\mathbf{x}_{i}-\mathbf{D} \boldsymbol{\alpha}_{i}\...
3
votes
1answer
61 views

Can one factorise a covariance matrix analytically or iteratively?

I have a covariance matrix which I would like to factorise. In more details, I would like to represent it in the following form: $ m \approx f \cdot f^T + diag(d^2), $ where $diag(d^2)$ means that I ...
2
votes
0answers
35 views

Expressing diagonal matrix using elementary matrices as generators in $\operatorname{SL}(\mathcal O_K \oplus \mathfrak a)$

Let $K$ be a real quadratic number field, $\mathcal O_K$ its ring of integers and an $\mathfrak a \subset K$ a fractional ideal. I've read in van der Geer that the group $$\operatorname{SL}(\mathcal ...
2
votes
0answers
26 views

Suitable matrix decomposition for incorporating diagonal offsets

The problem: I am faced with a problem where I need to repeatedly calculate: $$\left( {\bf A} + {\rm diag} \left( {\bf b} \right) \right)^{-1} {\bf c}$$ for a fixed, square, positive definite ${\bf A} ...
0
votes
0answers
32 views

Why does Householder tridiagonalization cost $4n^3/3$?

In Golub, Van Loan - Matrix Computations page 459, it says This algorithm requires 4n^3/3 flops when symmetry is exploited in calculating the rank- 2 update. But I don't quite understand that. How ...
0
votes
1answer
12 views

Can TDT^T transform a square matrix D into diag{D_0,0} where D_0 is nonsingular?

If $D\in R^{n\times n} $ is a square matrix with rank $m$ ($m < n$), can we always find a nonsingular matrix $T\in R^{n\times n}$ such that $$ TDT^T = \left[\begin{matrix}D_0 & 0\\ 0& 0\end{...
0
votes
4answers
93 views

Any nice way to calculate $A^n$

Let $A:=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$. I want to find a formula for $A^n$, is there any other way to do that than eigenvalue decomposition? I tried: $A^2 = \begin{pmatrix} 1 &...
2
votes
0answers
40 views

Finding generalized eigenvectors of a matrix

I would like to know how to find the generalized eigenvectors to the following matrix $A$, so that I can express $A$ as $PJP^{-1}$. $$ A = \begin{bmatrix} 1 & -3 & 1\\ 1 & 5 & -1\\2 &...
1
vote
1answer
24 views

Size of Jordan blocks according to the characteristic polynomial

Consider a Jordan matrix $\phi$ with the characteristic polynomial $$\chi_\phi(t) = \prod_{i=1}^m(t - \lambda_i)^{n_i}$$ where $\lambda_i \ne \lambda_j$ for $i \ne j$. I want to show that $n_i$ is the ...
0
votes
1answer
28 views

Eigen-decompostion involved with a semi-orthogonal matrix

Let $V$ be a $m$ by $n$ ($m>n$) semi-orthogonal matrix, i.e. $V^TV=I_n$ Can we say something about eigen-decomposition of $V^TPV$ where $P$ is a semi-positive definite matrix? Result would be ...
0
votes
0answers
35 views

Davis-Kahan Theorem for PSD Matrices.

Let $A=XX^T$ and $B=YY^T$ be symmetric, PSD matrices, such that $\lambda_k(A)-\lambda_{k+1}(A)$ is large. Suppose $Y=X+E$, where $\|E\|$ is small, and $X$ and $Y$ are not necessarily symmetric. Is ...
0
votes
0answers
14 views

Why is the objective of Matrix completion objective non convex?

Hi is there any general proof by taking second order convexity test to prove the following ? Why does the non-negative matrix factorization problem non-convex? I am self learning convex optimization ...
0
votes
1answer
25 views

Find the eigenvectors of ill-conditioned matrix

During numerical simulation of continuous time non-homogeneous Markov process, my propagator matrix becomes ill-conditioned, that is, the smallest eigenvalues exponentially decrease and become ...
0
votes
0answers
14 views

calculate or decompose a Fourier transform signal amplitudes with unknown weights on sources

I am trying to calculate , or approximate the solution of following Fourier-sine transform problem that can be expressed as a contributions of periodic sources $f_i(x)$ and weights $a_i(x)$ : $$F(k)...
1
vote
0answers
31 views

Matrix whose columns are the first $p$ left singular vectors [migrated]

Let's say I have a $m \times n$ matrix $A$. When I find its SVD, I get $p$ dominant singular values. How do I get the $m \times p$ matrix whose columns are the first $p$ left singular vectors of $A$?...
0
votes
0answers
9 views

Covariance of Wishart Distribution

Let us consider a random matrix $$W \sim Wishart(n, \Sigma_p),$$ i.e. $W \in \mathbb{R}^{p \times p}$ and $W = Z^T Z$ for $Z \in \mathbb{R}^{n \times p}$ and each row of $Z$ drawn iid from a $N(0, \...
0
votes
1answer
32 views

A question about positive definite matrix.

$\mathbf{A}$ is a real positive definite matrix. Show that there exists an upper triangular matrix $\mathbf{B}$ such that $\mathbf{A}=\mathbf{B}\mathbf{B}^{\mathrm{T}}$, and all the numbers on the ...
0
votes
2answers
27 views

How to solve xA = b using LU factorization [closed]

Given an N × N matrix A and its LU factorization LU = PA where L is lower-triangular and unit-diagonal, U is upper-triangular, and P is a permutation matrix. How to solve xA = b in O(N^2) flops?
0
votes
1answer
36 views

If I know the ODE of a PSD matrix, how can I find the ODE of its square root matrix?

I know that the ODE that describes how the covariance matrix $P$ changes over time is $$ \dot{P} = A P + P A^T $$ where both $A$ and $P$ are time-varying $n \times n$ matrices. Since $P$ is a ...
1
vote
1answer
24 views

Eigenvalues of a product of matrices with very large elements

Consider a set of matrices $M_i = \begin{bmatrix} e^{-b_i}(1+a_i) & a_i \\ a_i & e^{b_i} (1-a_i), \end{bmatrix}$ where $a_i$ and $b_i$ are of order $O(1)$. (This is an example of the type of ...
1
vote
1answer
43 views

Computation of Cholesky decomposition of Gram matrix from its components

Let's assume I have a tall matrix $\mathbf{X} \in \mathbb{C}^{m\times n}$, where $m \gg n$. I form the Gram matrix $\mathbf{A} = \mathbf{X}^*\mathbf{X}$, where $\mathbf{A} \in \mathbb{C}^{n\times n}$ ...
0
votes
0answers
30 views

For an $n \times n$ matrix whose SVD is $A = U D V^\top$, what do we know about $UV$?

I'm using a svd (singular value decomposition) function in a programming library that I didn't write. Given a square real nxn matrix, svd returns three values U,S,V where S is a vector designating a ...
0
votes
0answers
19 views

Lanczos algorithm for finding top eigenvalues of a matrix sum

Crossposted on Computational Science SE I am trying to find the top k leading eigenvalues of a NumPy matrix (using python dot product notation) ...
2
votes
0answers
16 views

$QR$ decomposition of block matrix

Given a square block matrix $$ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb R^{2d \times 2d}, $$ where $A, B, C, D \in \mathbb R^{d \times d}$. Is there some kind of a block $...
0
votes
0answers
8 views

Find Jacobian of the change of variables due to diagonalisation: $M=\mathcal{U} D_{M} \mathcal{U}^{T}$

Let $M$ be a (square) real symmetric matrix. I can decompose $M$ in terms of its eigenvectors $\vec{v}$ and eigenvalues $\lambda$: $$M=U D_{M} U^{T}$$ Where $U$ is an orthogonal matrix composed of the ...
0
votes
0answers
29 views

How many matrices over two element field are nilpotent of degree smaller equal 3.

my post is related with 1 page Shitov's paper: https://www.researchgate.net/publication/332630367_The_ring_mathrmM_8k4mathbbZ_2_is_nil-clean_of_index_four/link/5ceea7d292851c4dd01a403c/download Maybe ...
0
votes
0answers
15 views

Maximizing correct(or incorrect) labeling and prove its bound.

Suppose there are M workers labeling N items (items have true label +1/-1). For each worker, it has $\frac{1+p_i}{2}$ $(-1\le p_i\le 1)$ successful probability to recover the true label of an item. ...
1
vote
0answers
67 views

How to Cast a Quadratic Constraint as Convex if Q is a (PSD) Variable?

Background I am working on a stochastic optimal control problem and $x$ represents the state of the dynamic system. The optimal state must be found, but, due to the stochastic nature of the problem, ...
2
votes
1answer
78 views

Matrix function decomposition of the form $A(\theta)=B(\theta)^\top Q B(\theta)$

Suppose I have a matrix function $A:\mathbb{R}^n\to\mathbb{R}^{n\times n}$, for which I know the following properties hold: $A(\theta)$ is real, symmetric and bounded for all $\theta$ $A(\theta)$ is ...
1
vote
0answers
59 views

How I can separate a matrix that is not positive definite, into two matrices?

I want to separate a matrix that is not positive definite, into two matrices in Matlab like this: $$ Q=S^{T}S $$ for example this matrix: $$ Q=\Biggm[\matrix{-92.316 &31.78&240.417\cr 31.78 &...
1
vote
1answer
43 views

Schur decomposition nonnegative real numbers on the diagonal

Is it possible to have a Schur decomposition of a matrix $A=URU^H$ so that the upper triangular matrix $R$ only has real non-negative numbers on the diagonal? I realize the diagonal of $R$ is ...
2
votes
4answers
120 views

QR factorization: $A=QR$ and $R=Q^TA$ gives $Q Q^T = I$

I am learning linear algebra and I am on QR factorization: $A = QR$. A solution to $R$ is $R = Q^TA$, where $Q$ and $A$ have size $m * n$ and $R$ is $n * n$, which means that $Q$ is not necessarily ...

1
2 3 4 5
43