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Questions tagged [matrix-decomposition]

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

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164 views

Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
6
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0answers
559 views

Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?

I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has ...
5
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0answers
98 views

Fast arbitrary decomposition of a positive-definite matrix

Given a positive-definite $n\times n$ matrix $\mathbf{A}$, my goal is to present it as a product of the form $\mathbf{H^TH}$, where $\mathbf{H}$ is an arbitrary $n\times n$ matrix. Cholesky ...
4
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37 views

Sufficient/necessary condition for submatrix determinant (minor) that decreases with size?

Definition. Given a square matrix ${\bf{A}}=[a_{ij}] \in {\mathbb{C}^{n \times n}}$, the submatrix ${{\bf{A}}_{{i_1},{i_2},...{i_k}}}$ is formed by retaining the $({i_1},{i_2},...{i_k})$-th rows and ...
4
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4k views

Checking positive semidefiniteness in MATLAB

Let $\mathbf{A}$ be a $n\times n$ matrix. I want to check in MATLAB if it is PSD or not. Which tests, in MATLAB, should I do for this purpose? I know that if $\mathbf{A}$ is PSD then following holds ...
4
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171 views

Is there a decomposition $U U^T$?

We know that there exist the Choleski decomposition $ M = L L^T $ where $M$ is a positive definite matrix and $L$ a lower triangular one. Does it exist a similar decomposition in $ M = U U^T$ ...
4
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83 views

To what extent are the Jordan-Chevalley and Levi Decompositions compatible.

I know that the Jordan-Chevalley decomposition for real Lie algebras only applies to semisimple Lie algebras, but in general the addititive J-C decomposition says that for ANY operator, we can ...
3
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48 views

Find rank of the special block symmetric and persymmetric matrix

I meet a difficult problem recently. The problem is to find the rank of a special matrix: $$X = \begin{bmatrix} S & P \\ P & JSJ\end{bmatrix}\in \mathbb{R}^{2m\times2m} ,$$ where $S$ is ...
3
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30 views

How to find the inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ without forming the Kronecker product?

Is there a good way to compute the inverse of Inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ that doesn't require forming the full Kronecker product? Here $A$ is symmetric, positive ...
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56 views

Papers on Tensor factorization

I started reading a few papers on Tensors. Right now I am reading a paper by Chi and Kolda called “On Tensors, Sparsity, and Nonnegative Factorizations”. I passed linear algebra and calculus courses. ...
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36 views

A set of four matrix equations

Let $\{\mathbf b_1,\mathbf b_2,\mathbf b_3\}$ be a basis for $\mathbb R^3$ and consider three arbitrary vectors $\mathbf w_i=\sum_j\omega_{ij}\mathbf b_j$. We define the following two $3\times 9$ ...
3
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84 views

Characterize set of matrices that are orthogonal in two particular senses

Does there exist an analytical characterization of the set of matrices $\Gamma_k\in\mathbb{R}^{m\times n}$ such that both $$ \sum_{k=1}^K\Gamma_k^T\Lambda_0\Gamma_k=I $$ and $$ \sum_{k=1}^K\Gamma_k\...
3
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0answers
88 views

Is a principal submatrix of a diagonalizable matrix diagonalizable?

Let $A$ be diagonalizable, i.e., $A=X \Lambda X^{-1}$ for some diagonal matrix $\Lambda$. Consider $B$ which is a principal submatrix of $A$. Does there exist an invertible matrix $Y$ and a diagonal ...
3
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245 views

Formulas for the entries of a matrix

$$A = \begin{pmatrix}2&2\\ -4&8\end{pmatrix}$$ Find formulas for the entries of $A^n$, where $n$ is a positive integer. I think I know the method to solve this problem but I am only ...
3
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0answers
80 views

Factorize product of matrices without actually computing the product of matrices

I have a a system of equations $$ (B^TB + C)x = B^Tb $$ Here, $B$ is a very large, dense matrix and $C$ is a very sparse symmetric matrix. Computing $B^TB$ is infeasible (although computing $BB^T$ ...
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77 views

Cholesky Algorithm example

I am trying to understand the exact form of the Cholesky algorithm used in the following exercise: Question: Find all values of $\rho$ such that the following matrix is symmetric positive ...
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487 views

What is the Householder matrix for complex vector space?

For $\Bbb R^n$, Householder matrix $Q=I-2vv^T$ is an operator that maps a vector to its reflection across a hyperplane of normal $v$. The following is an illustration for Householder operator of a ...
3
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119 views

How to factorize (decompose) the sum of matrix powers and its transpose?

Suppose I have a matrix $M$, such that: $$M = \sum_{k=0}^{\infty} c^{k}(A^{k})^{\top}DA^{k} = D + cA^{\top}DA + c^{2}A^{2\top}DA^{2} + \cdots,$$ where $c \in [0,1]$ is a scalar, D is a diagonal ...
3
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0answers
68 views

Inverse of matrix of a specific form.

I want to compute a matrix of the form $$(D+AA^{\prime})^{-1} AA^{\prime} \tag{1}$$ where $D$ is a diagonal matrix and $A$ is a $p \times k$ matrix with $p \gg k$. I got this expression after ...
3
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212 views

Singular value decomposition of a low rank weak diagonally dominant M-matrix. When is the unitary polar matrix positive semi-definite?

Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...
3
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0answers
126 views

PCA derivation by minimizing reconstrution squared error plus orthonormality constraints

I'm trying to better understand the link between PCA and Matrix Factorization of the form $X \approx WH$. I've read somewhere that the PCA solution can be also derived from the following cost ...
3
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378 views

Connection between SVD and Discrete Fourier Transform for Denoising

Denoising signals (in particular, 2D arrays, such as images) can be done by removing the high frequency components of the discrete Fourier transform (which is related to convolution with a Gaussian ...
3
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0answers
1k views

QR factorization: Givens rotations vs. Householder reflections

Here is the algorithm for Housholder $QR$ factorization: $A$ is a $m \times n$ matrix and $m$ is greater than $n$. $For \quad k=1 \quad to \quad n$ $\quad x = A_{k:m,k}$ $\quad v_{k} = sign(x_{1})|...
3
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0answers
180 views

How to improve/increase full rank of a matrix

I have a matrix $W$ which can be decomposed as $$ W=\left[ \begin{array}{c|c} U & 0\\ \hline Y&Z \end{array} \right] $$ In which, $U,Z$ are given and have $rank(U)=...
3
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0answers
6k views

Decompose 3D rotation matrix into rotation around x, y and z-axis

I have a rotation matrix R, that produces an arbitrary rotation in a 3D space. I would like to decompose it into 3 rotation matrices Rx, Ry and Rz so I can use and apply only xy in plane rotation (...
3
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0answers
159 views

Decomposition of a matrix into two non-square matrices

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
2
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0answers
19 views

how to decompose matrix like this?

I want to have two matrixes of $ A $ and $B$ of a certain size, multiply them together $C = AB$, do some operations on $C$ to make $C_2$ and then decompose the matrix into to matrices of the same size ...
2
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0answers
20 views

Is there some name for a matrix which is unitarily similar to its negative?

I have a code spitting out matrices $A$. I am trying to understand the structure of these matrices. I have identified that we can always unitarily transform the matrix $A$ to $-A$ via a transform $A'=...
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0answers
47 views

Is there a closed-form formula for the derivative of the positive factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
2
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0answers
36 views

SVD of a specific matrix and Singular values behaviour

I have a rectangular matrix $A \in \mathcal{M}_{l,n} (\mathbb{C})$, $l>n$ which has this property : $$A=\left[\begin{matrix} M_1 & i M_2 \\ M_2 & -i M_1 \end{matrix}\right]$$ Where $M_i$ ...
2
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0answers
57 views

Subgroup generated by union of two maximal compact subgroups of $GL_2(\mathbb{Q}_p)$

Let us denote by $G:= GL_2(\mathbb{Q}_p), G_0:= GL_2(\mathbb{Z}_p), g:= \begin{bmatrix}0 & 1\\p& 0\end{bmatrix}$ and by $G_1:= g G_0 g^{-1}$. I want to know if we have a good description of ...
2
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0answers
36 views

Real matrix decompose into symmetric matrices

Using Jordan standard form, we can prove Any complex matrix can be decomposed into the product of two symmetric matrices, and one of them is invertible. then how to prove that Any real square ...
2
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0answers
32 views

Schur factorization from spectral decomposition

Let $A \in \mathbb C^{d \times d}$. The Schur factorization of $A$ is the decomposition $A = U T U^\ast$, where $T \in \mathbb C^{d \times d}$ is an upper triangular matrix and $U \in \mathbb C^{d \...
2
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0answers
27 views

Obtain unknown matrix from eigendecomposition.

I have doubt with the mathematical tools used in a problem of Signal Analysis: I have a complex observable series $Y(t)$ which is the result of summing two complex r.v $X(t)$ (unobservable) and a $\...
2
votes
0answers
71 views

Optimization over a complex unit circle

I have a problem in the following form: $\underset{\mathbf{x}}{\text{min}} \mathbf{||Ax - b||^2_2}$ s.t $|[\mathbf{x}]_n| = 1; \forall n$ $\mathbf{A}$ is a complex $M \times N$ matrix and $\mathbf{...
2
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0answers
46 views

A decomposition for triangular matrices with positive diagonal entries

Let $A\in\mathbb{R}^{n\times n}$ be an upper triangular matrix with strictly positive diagonal entries. Is it possible to find: an orthogonal matrix $T\in\mathbb{R}^{n\times n}$ ($T^\top T=...
2
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0answers
37 views

Name of orthogonal/unitary matrix decomposition?

Suppose we have an orthogonal/unitary matrix $T$ of even dimension. Then we can decompose it into: $$ T = \begin{bmatrix} U_1 & 0 \\ 0 & U_2 \end{bmatrix} \begin{bmatrix} R & -\sqrt{...
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0answers
11 views

Volume of subgroups and coset decompositions

Let $F$ be a number field, $\mathfrak{o}$ be its ring of integers and $\mathfrak{p}$ be the associated maximal ideal. Consider the subgroup $H$ of ${GL}(3, \mathfrak{o})$ given by matrices of the ...
2
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0answers
92 views

Generate approximately semi-orthogonal tall matrix approximately satisfying constraints

I have a set of matrices $\{(A_i,D_i)\}$ for $i\in\{1,\ldots,n\}$, where: Each $D_j\in\mathbb{R}^{S\times S}$ is diagonal, and every entry on the main diagonal is non-negative. Each $A_j\in\mathbb{R}^...
2
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0answers
43 views

A blockwise Matrix transpose

I currently have a problem, that can be stated like this: Given two matrices $ A= \begin{bmatrix} A_1 \\ A_2\end{bmatrix}$ and $B=\begin{bmatrix} B_1 &B_2 &B_3\end{bmatrix}$, we basically ...
2
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0answers
93 views

Singular value decomposition of Block Diagonal matrix.

I have numerically seen the SVD of a bock diagonal matrix to be equal to the SVDs of individual 'elements' stacked. That is the matrices U and V would again be block diagonal with the U and V of ...
2
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0answers
65 views

Decompose stochastic matrix in product of two stochastic matrices

There exist stochastic matrices $Q$ such that there is no stochastic matrix $P$ such that $P^2=Q$. I am interested in the following problem: For a given stochastic matrix $Q$, find stochastic ...
2
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0answers
33 views

Matrix decomposition to its original product

If I have a squared symmetric matrix $X(m,m)$ which is a result of the product: $X =Y^T A Y$, in which $Y(n,m)$; and $A(n,n)$ is a diagonal matrix. My question is: Is it possible to get the original $...
2
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0answers
74 views

Singular Value Decomposition for Rectangular Matrices

In the book Linear Algebra Done Right (Axler, 237), the following is the statement of the theorem that gives us the existence of a singular value decomposition: 7.51 $\qquad$ Singular Value ...
2
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0answers
25 views

How to deblur a image matrix blured by two circulant matrix?

We suppose an image matrix $X\in \mathbb{R}^{n_1\times n_2}$ is blurred by two circulant matrices $\Phi_1 \in \mathbb{R}^{n_1\times n_1},\Phi_2 \in \mathbb{R}^{n_2\times n_2}$. We can observe the ...
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0answers
58 views

Simplify $x^TA(AA^T+I)^{-1}A^Tx$

I need to simplify the term $$x^TA(AA^T+I)^{-1}A^Tx$$ where $A$ is $N\times n$ matrix and $x$ is a $N\times 1$ vector. I have some information of $x^Tx$. Could anyone know how to simplify the term ...
2
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0answers
241 views

Show that partial pivoting leads to an $LU$ decomposition of $PA$.

Exercise: show that for every non-singular matrix $A$, partial pivoting leads to an $LU$ decomposition of $PA$ so: $PA = LU$. I have the following theorems I can use: Theorem 1: Assume that the ...
2
votes
0answers
158 views

Prove using Schur decomposition $\lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1$

a) Prove using Schur decomposition $\lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1$ where A is in $\mathbb{C}^{mxm}$ and $p$ is the spectral radius. b) $\lim_{t \rightarrow \infty}||e^{At}||=0 \...
2
votes
0answers
137 views

Showing infinity norm of diagonally dominant matrix is less than 1

We have a matrix A that is diagonally dominant by rows with decomposition $A = D - L - U$. The matrix $T = D^{-1}(L+U)$ and I want to show that $\lVert T\rVert_\infty < 1$ is always true. I know ...
2
votes
0answers
34 views

SVD for collaborative filtering when number of latent concepts equals the rank

Consider we perform collaborative filtering on a rating matrix $A$ of rank $r$, where we assume that all the unobserved entries (items that haven't been rated per each user) are marked by zero. If ...