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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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Detect linearly dependent columns from a full-row rank matrix.

Let $A\in \mathbb{R}^{m\times n}$, with $m\leq n$, be a full-row-rank matrix. Then, there exist a collection of $n-m$ columns in $A$ that are linearly dependent on the other columns. What is the most ...
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2x2 blocks in the QZ algorithm

How are the $2\times2$ blocks supposed to be diagonalized in the QZ-Algorithm? Taking the matrix pencil (A,B) and finding it's generalized Hessenberg decomposition (H,R) for which $\exists Q,Z \in \...
Littlejacob2603's user avatar
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Matlab qz algorithm not reliable

I programmed my own version of the qz algorithm and, while testing it's results using the matlab qz algorithm, I found a particular case where my solution reaches an upper-triangular matrix and matlab ...
Littlejacob2603's user avatar
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Is there a closed form solution?

Question: Let $A\in\mathrm{GL}(n,\mathbb{R})[[u]]$ with $A=A^T$, that is a symmetric real matrix, which can not be decomposed into a blockmatrix with $0$ blocks, such that $$ A = \begin{pmatrix} A_1 &...
dForga's user avatar
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Degrees of freedom of an $r$-ranked tensor?

I'm trying to determine the degrees of freedom for parameters of a tensor in shape $(J_1,\dots,J_D)$ and with rank $r$, where "rank" refers to the smallest number of rank-1 tensors whose sum ...
graphitump's user avatar
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Inverse and Determinant of Matrix $Axx^TA+cA$

Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
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5 votes
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If I can solve $Ax = b$ efficiently, is there a method to solve $(I+A)x=b$ efficiently?

Say if I already have the LU factorization of a square matrix $A$, is there an efficient way to get the LU factorization of $I+A$? (We may assume all the matrix I mentioned is invertible.) I know from ...
qdmj's user avatar
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Best method for sequential small size Hermitian smallest eigenpair problem

I am working on a perhaps rather strange problem. To find the smallest eigenvalue and its eigenvector, for a large number (a few billions) of small (20 * 20 to 200 * 200) Hermitian matrices. These ...
Scriabin's user avatar
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1 answer
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If $A^*A=A^*B=B^*A=B^*B$, prove that $A=B$.

Suppose that $H$ is a Hilbert space and $A,B\in B(H)$ are such that $A^*A=A^*B=B^*A=B^*B$,then $A=B$.I'm having difficulty trying to read a proof that solves this problem. First, the author decomposes ...
OSCAR's user avatar
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Fast way to compute the largest eigenvector of an expensive-to-compute matrix

Consider an $N \times N$ Hermitian positive semidefinite matrix $M$. Computing the elements of $M$ is expensive so we wish to compute as few as possible. We can assume that $M$ can be approximated as ...
Sah20000000000000's user avatar
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4 answers
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$O$ orthogonal with $\det(O)=-1$ implies $||\Omega - O \Omega O^{T}|| = 2 $?

Let $O\in \mathrm{O}(2n)$ be an orthogonal matrix. Let $\Omega$ be the matrix $\Omega:= \bigoplus^n_{i=1} \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}.$ Is it true that: $\det(O)=-1$ ...
Dante Perès 's user avatar
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Quadratic form of a matrix where non-standard decomposition is known

Let $1\le m<<n$ for integers $m,n$. I have two symmetric matrices of size $n\times n$, say $A$ and $B$. My goal here is to know a simple form, or an upper bound on the following quadratic form: $...
2019ys's user avatar
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Minimize $||A-AWW^TA^T||_F$ w.r.t. $W$

Given $n \in \mathbb{N}$ and $A \in \{0,1\}^{n \times n}$, we aim to find $$\arg \min_{W \in \mathbb{R}^{n \times n}} f(W) = ||A-AWW^TA^T||_F,$$ where $||\cdot||_F$ represents the Frobenius norm with $...
Vezen BU's user avatar
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Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.

The problem is a conituation of this problem, but over finite fields. In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
X.H. Yue's user avatar
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Euclidean norm of a vector resulting from a matrix multiplied by an orthonormal matrix multiplied by an arbitrary vector!

Let $A \in \{0,1\}^{m\times n}$ be a binary matrix with $ m < n$, and let $U \in \mathbb{C}^{n \times n}$ be an arbitrary orthonormal matrix. Let $\sigma_{min}$ denote the smallest non-zero ...
Drimitive Watson's user avatar
4 votes
1 answer
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Binary matrix power for a specific entry.

Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$. I am interested in the following decision procedure: Given a ...
Bader Abu Radi's user avatar
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Understanding QR decomposition in the context of a cubature Kalman filter

I am working on implementing a square-root cubature kalman filter based on the algorithm presented in this conference paper, as well as this paper more broadly. I have got the algorithm to run; ...
NorthwoodsEngineer's user avatar
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Finding Correspondence between Matrices after Decomposing a Hermitian Matrix

After decomposing the Hermitian matrix $M$, I obtain a set of matrices $D_j$, where each $D_j$ is defined as $D_j = \lambda_j u_j u_j^H$, $\lambda_j$ and $u_j$ are eigenvalues and eigenvectors, and $M$...
Cyberturist's user avatar
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Deduce $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix}$ or the simple parity-check matrix just from the final matrix-vector product equations

A linear combination like $A_{1}+ A_{2}$ serves as a backup of $A_{1}$ when $A_{2}$ is known, and serves as a backup of $A_{2}$ when $A_{1}$ is known. As a result of linearity, any two out of $A_{1}x$,...
triple_tactic's user avatar
1 vote
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How to compute $RA$ in $O(n^2)$ operations instead of $O(n^3)$ using Householder reflection

I am currently writing a program that performs QR decomposition on a matrix $A$. The guidelines to my assignment tell me that once I calculate $R$ using Householder reflections, there is a way to ...
Amrutha Paval's user avatar
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Projection of vector

The projection of a vector $x$ onto a vector $u$ is $proj_u(x) =\frac{\langle x, u \rangle}{\langle u, u \rangle}u.$ Projection onto $u$ is given by matrix multiplication $proj_u(x)=Px$ where $P=\frac{...
David's user avatar
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Symplectic approximation to a given matrix

I am interested in understanding methods to rigorously compute the closest symplectic matrix approximation to an arbitrary matrix $A$. A symplectic matrix $S$ satisfies the condition $ S^T J S = J $, ...
Dante Perès 's user avatar
1 vote
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Interpretation of QR "values" (a la singular values)?

Let $A\in\mathbb{R}^{m\times n}$ with $m>n$ and $\text{rank}A=n$. There exists $\hat{Q}\in\mathbb{R}^{m\times n}$ with orthonormal columns and $\hat{R}\in\mathbb{R}^{n\times n}$ upper triangular ...
2016applicant's user avatar
2 votes
1 answer
19 views

Equation for Counting Unique RREF "Cases" for any (m x n) Matrix

Given a real matrix $A$ of size $(m \times n)$, I am seeking to determine the number of unique reduced row echelon form (RREF) "Cases" that exist for a given matrix size. Two RREF matrices ...
FaffyWaffles's user avatar
1 vote
1 answer
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How to derive the relation about Jordan decomposition of a matrix?

Assume that $v$ is an eigenvector with an eigenvalue of $0$ in matrix $H$, and its Jordan decomposition $H^J=SHS^{-1}$ satisfies $$ H^J=\left( \begin{matrix} 0& 1& \cdots& 0\\ 0& ...
SHBooKP's user avatar
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Distance between subspaces with spectral norm

I was trying to prove this following theorem , Let $$ W=\begin{bmatrix} \underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2} \end{bmatrix} $$ $...
Kaustubh Limaye's user avatar
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Norm inequality for eigendecomposition and oblique projectors

Consider a stochastic matrix R which permits an eigendecomposition into oblique projectors, $$R = \sum_{\lambda} \lambda C_{\lambda}$$ I've observed the following 2-norm inequality in a number of $R$ ...
Renmusxd's user avatar
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Inverting a specific symmetric matrix preserves its zero entries

Suppose $S$ is an invertible symmetric matrix with the following property: For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number ...
Yujian's user avatar
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Is polar decomposition commutative for diagonal matrices?

I did a error while understanding about the polar decompositon. I thought polar decomposition is PU, but it is UP. While trying ...
Manu's user avatar
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Finding the eigenvalues of a tridiagonal block matrix of special form

Consider the following symmetric tridiagonal block matrix: $$\begin{bmatrix} 2I_{N \times N} & -I_{N \times N} & O & \dots & O &O \\ -I_{N \times N} & 2I_{N \times N} &...
Stack_Underflow's user avatar
5 votes
1 answer
201 views

In what sense are similar matrices "the same," and how can this be generalized?

I sort of intuitively see why we care about similar matrices, i.e., when $A=S^{-1}BS$ for some invertible matrix $S$. But I want to make this intuition more precise and abstract. Matrices: First of ...
WillG's user avatar
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1 answer
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Inverse $T$ matrix of a 3*3 matrix. [closed]

I have the matrix $$ A= \begin{bmatrix} 1 & 4 & 1\\ 0 & 2 & 5\\ 0 & 0 & 5 \end{bmatrix}. $$ I have found the $$ T= \begin{bmatrix} 1 & 4 & 1\\ 0 & 0 & 1\\ 0 &...
Irini's user avatar
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1 vote
1 answer
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Orthogonal procrustes with kernel constraint

Given a matrix $M \in \mathbb{R}^{n \times m}$ with $n > m$ and an arbitrary vector $v \in \mathbb{R}^n$, I am looking for an analytical solution for the orthogonal matrix $R \in \mathbb{R}^{n \...
tommym's user avatar
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1 vote
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Spectrum of compact self-adjoint operator vs symmetric matrix

I know that compact self-adjoint operators have a countable, orthonornal eigendecomposition, and the spectrum is real and positive, and the only cluster point is zero. I know that self-adjoint ...
900edges's user avatar
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3 votes
1 answer
91 views

Solve $\| X A - B \|$ subject to $X C = C X$

Given $A, B \in \mathbb{R}^{n \times k}$ and S.P.D. $C \in \mathbb{R}^{n \times n}$, I would like to find an analytical solution for the matrix $X \in \mathbb{R}^{n \times n}$ that minimizes \begin{...
tommym's user avatar
  • 371
2 votes
0 answers
9 views

How to describe the partition of $GL_3(\mathbb{C})$ by Bruhat decomposition accurately?

We know that $GL_n(\mathbb{C})$ can be decomposed as $GL_n(\mathbb{C}) = \bigsqcup_{w \in W}BwB$ where $B$ is the subgroup of upper triangular invertible matrices and $W$ is the Weyl group isomorphic ...
YSouSerious's user avatar
2 votes
1 answer
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Relation between eigenvalues of a general complex matrices and its realed one

Let $A=A_R+iA_I$ be a $n\times n$ complex matrix. If we want to solve a linear system with regard to $A$ and do not want to take complex arithmetic, then we often generate the following real matrix: \...
Adrain's user avatar
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QR algorithm fails to converge (bad shift?)

Problem: My code-based implementation of the implicit QR algorithm fails to converge for certain special cases, and it's because those cases have bad shift values. What are those special cases: While ...
Math Machine's user avatar
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How to show that the Gaussian Process parameters decreases with more training data

The posterior mean prediction of a Gaussian Process is given by $$\mu(x_*) = \sum_{i=1}^n\alpha_i k(\mathbf{x}_i,\mathbf{x}_*) $$ where $$\alpha = (K + \sigma_n^2I)^{-1} \mathbf{y}$$ Can we show that $...
Tee's user avatar
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1 vote
2 answers
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What is a complete geometric interpretation of the eigendecomposition of matrices?

For reference, eigendecomposition of a matrix $A$ $\in R^{n \times n}$ is defined as: $A = P \Lambda P^{-1}$ where $P$ is a matrix whose columns are the eigenvectors of $A$, and $\Lambda$ is a ...
Arbaaz Qureshi's user avatar
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The order of decomposition of matrices

In QR decomposition of a matrix, Q represents the orthogonal matrix and R represent upper triangular. I was wondering if a matrix could be decomposed into orthogonal and lower triangular and found LQ ...
Jay's user avatar
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1 answer
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When complete pivoting fails for a linear system

As I understand it, the following three pivoting techniques: Partial pivoting (which exchanges rows determined by a sub-column search) Rook pivoting (which exchanges rows or columns based on sub-row ...
Attack68's user avatar
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find $D$ of $(D+A)$ for $diag((D+A)^-1)=k$

I am wondering whether it is possible to derive $D$ for $diag((D+A)^{-1})=k$ where $diag()$ produces a vector of diagonal elements of a squared matrix, $D$ is an unknown diagonal matrix with possible ...
user1407220's user avatar
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Solving Matrix Equation using SVD

I'm reading this paper by Bishop and Tipping. They solve the equation $$(SC^{-1} - I)W = 0$$ Where $W \in \mathbb{R}^{d \times q}$ and $S , C \in \mathbb{R}^{d \times d}$ and $C = WW^T + \sigma^2 I$ ...
Harry's user avatar
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1 vote
0 answers
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Estimating the parameters of an ellipse

Problem definition Consider a dataset composed by $m$ bivariate measurements \begin{equation*} y_j \sim \mathcal{U}(\mathcal{E}(\ell_1,\ell_2,\theta)) \qquad j=1,2,\dots,m \end{equation*} uniformly ...
matteogost's user avatar
3 votes
3 answers
187 views

The relation between the eigenvalue of a Hermitian matrix and the block matrix that composed by it real and imaginary part

Recently I am reading a paper. In their "Proof of Lemma 1" on page 24, they have: $$\lambda_+(\mathbf{Q})=2\lambda_+(\tilde{\mathbf{Q}})$$ where $\mathbf{Q}$ is a Hermtian matrix, $\tilde{\...
tyrela's user avatar
  • 353
0 votes
1 answer
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Specific basis for the space of symmetric matrices

Consider the space of symmetric matrices $symm(M)$ over reals of dimension $n \times n$. It is clear that there is a straightforward basis for this space where for any $i \ge j$ $M_{ij}(m,n) = 1$ if $...
supernova's user avatar
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Factorizing $AMA^T+N=WW^T$ efficiently.

This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed. ...
P. Quinton's user avatar
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3 votes
1 answer
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$(A+B)^2\|B^{-1}\|-A$ is positive definite?

If $A$ and $B$ are positive definite matrix, $(A+B)^2\|B^{-1}\|-A$ is also positive definite? When $A$ and $B$ are real numbers, then $(A+B)^2\|B^{-1}\|-A$ is obviously positive definite. When $A$ and ...
zeng's user avatar
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Differences between QR and LQ matrix decomposition

I'm supposed to do a variety of matlab experiments for different factorizations and two of those are QR and LQ factorizations, which for matrix $A$ ${m \times n}$ yield $$A=Q[R_1\;R_2]$$ where $R_1$ ...
Weyr124's user avatar
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