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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Finding the maximal direct sum decomposition of a linear map

Suppose a matrix $M$ over some field $k$ allows a diagonalisation $P D P^{-1}$. On the level of linear maps, this is a decomposition of the linear map $f: V \to V$ represented by $M$ into $$V \cong \...
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How to relate spectral radius and Cauchy interlace theorem in a separate equation

Consider a a matrix $A_{n\times n}$. Also consider another matrix $M_{(n-1)\times(n-1)}$ which is obtained by deleting the $i^{th}$ row and column of $A$. I have the following equation in $M$ \begin{...
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Any linear transformation is a sum of a rotation and streching. So Is it possible to decompose any $A$ real matrix into such $R$ and $S$ components?

The Question Let $A \in \mathbb{R}^{n \times n}$ be the matrix of a linear transformation. I have learned that any linear transformation is either a rotation, a streching, or a mirroring (with the ...
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Approximating the eigenvalues and eigenvectors of $B := A + E$

Suppose I have a matrix $B := A + E$, where $A$ is diagonal and $E$ is an off-diagonal, symmetric matrix whose non-diagonal elements are small. Is there any way to obtain the approximate eigenvalue ...
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Matrix powers up to multiplicative factor

Let $A$ be a real $n\times n$ matrix, $A_n = A^n$, and $$ \bar A_n = \lbrace\alpha A_n, \alpha\in \mathbb{R}\rbrace.$$ I am interested in characterizing the behavior of $\bar A_n$ when $n\rightarrow \...
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Recommendation on linear operator decompositions

Looking for a recommendation on theory of linear operator decompositions. I know some basics of linear algebra: what is a vector space, what is a linear operator, how to compute eigenvalues/...
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Matlab.Block matrices

I have implemented algorithm of Crouts method. But I don't have any idea how to create this M function in Matlab and implement in my algorithm .Please help me. CODE of algorithm: function [L,U] = ...
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Decomposition of a 4D rotation into a particular sequence of simple rotations.

It is known that any rotation in $SO(n)$ can be decomposed into a particular sequence of $n(n-1)/2$ simple rotations (that is, rotations which rotate a 2D plane in $\mathbb{E}^n$). The procedure to ...
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Any thought on that rank similar object?

Context : This problem arises on some exploration around Wyner's common information in information theory and the related minimization problem. Problem : Let $A$ be a $m\times n$ real matrix. Its ...
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If $𝑓∘𝑔∘ℎ=𝑓 ∧ 𝑔∘ℎ∘𝑓=𝑔$ then must $ℎ∘𝑓∘𝑔=ℎ$?

If not, then What can be said of each $𝑓,𝑔,ℎ$ and are there any simpy-definable minimal conditions imposable upon one or more of the indexable functions that would ensure this symmetric closure? ...
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LU decomposition with partial pivoting

Different sources* all introduce LU with partial pivoting with multiplying matrix $A$ by permutation matrices $P_i$ and matrices used in classical LU decomposition $L_i$ such that $$ L_nP_n \cdots L_1 ...
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Transform stacked matrix into block-diagonal form

Consider two matrices $A$ and $B$ that get stacked to form a (tall) matrix $J$, $$ J = \left[\begin{array}{l} A\\ B \end{array} \right]. $$ Assume that $\text{rank}(J) = \text{rank}(A) + \text{rank}(B)...
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Are the determinants of a matrix and the diagonal matrix obtained after diagonalization equal? [closed]

This question is related to a derivation step needed to find an n-dimensional generalization of the Gaussian integral, derived here: reference for multidimensional gaussian integral Is it true that ...
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Principal matrix square root of block matrix

I want to compute the principal square root $C=C^{1/2}C^{1/2}$ where the matrix $C$ is symmetric and positive definite (SPD). Let $$ C := \begin{bmatrix} A & B^T\\ B & D \\...
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Why are Latent Factor Models called "SVD"?

Latent factor models (usually used for recommender systems) are a matrix decomposition of matrix $R$ such that $$ R = P \cdot I^T $$ with the "twist" that values in $R$ can be missing (which ...
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Polar decomposition and transpose similarity

Is there any manner to prove that an inversible real matrix is conjugate to its transpose using the polar decomposition ? This would be, in this particular case, a much more simpler proof than using ...
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congruent matrices and eigenvalues

Let $Y$ be diaogalizable matrix $Y = PDP^\top$, where $P$ is orthogonal and $X$ be congruent to $Y$, that is there exists an invertible matrix $A$ such that $X=A^\top YA$. Thus we can find a linearly ...
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Constructing a special kind of SVD

Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as $$ A = XD_AY^H \\ B = XD_BY^T $$ where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
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Time complexity of getting $r$-largest eigenvalues and vectors of a symmetric matrix.

$A$ is a $n \times n$ symmetric matrix. I would like to know the time complexity of calculating $r$-largest eigenvalues and vectors. When we need all eigenvalues and eigenvectors, it means $r=n$, I ...
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Relationship between the last column of Q and the last row of L inverse in QR & PLU decomposition

I am trying to figure out the relationship between the last column of Q in QR decomposition (A = QR) and the last row of L inverse in PLU decomposition (PA = LU), but I am stuck because of P. PA = LU -...
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Computing inverse elements of symbolic matrices with binary variables

I'm working with symmetric, symbolic matrices $A$ with real coefficients and linear binary variables like $$ A = \begin{pmatrix} 0.5x_0 & 0.3x_1+0.002x_2 & 0 & 0 \\ 0....
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Diagonalizing of a block matrix which satisfies some conditions

Let $A$ be a matrix over a division ring $D$. Assume that $A$ can be written as $\left(\begin{smallmatrix}A_{11}&A_{12}\\0&I\end{smallmatrix}\right)$ such that $A^2=I$. Then I wonder if $A$ is ...
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Uniqueness of monic polynomial that can achieving $\inf_{p_n \in P_n} ||p_n(A)||>0$ with $A$ being an nonsingular matrix.

Problem: Let $A$ be an $m \times m$ nonsingular matrix. Suppose $\inf_{p_n \in P_n} ||p_n(A)|| = ||p^*(A)|| > 0$ where $P_n$ denotes the set of all degree-n monic polynomials: $$P_n ={p:p\text{is a ...
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Clarifying the "why" behind SVD, LU, QR and Woodbury matrix identity decompositions

I've been reading about matrix decompositions for a while now, but I always somehow seem to forget the reasons why they exist, and this irritates me when I see them applied in various branches of ...
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A notation $H$ on top of matrices

Could anyone tell me what the $H$ notation generally used on the top of some matrices is meant to be in this paper by Schmidt? The first usage is in equation 2.9 on page 9. Does it mean to be ...
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For any matrix $A \in M_{n\times n}(\mathbb {R})$, there exists a permutation matrix $P$ such that $PA=LU$.

For any matrix $A \in M_{n\times n}(\mathbb {R})$, there exists a permutation matrix $P$ such that $PA=LU$.Here P is defined as a matrix resulting from any number of row interchanges in the $I_{n\...
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Matrix whose element $A_{m,n}$ only depend on $m - n$ or $m+n$

I am encountering a problem that would require me to look into matrices with the following form: the elements $A_{m,n}$ only depend on $m-n$ or $m+n,$ such as: $$ A = \begin{bmatrix} a&b&c\\ ...
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Deriving SVD from polar decomposition

The Wikipedia article on the polar decomposition states that, for any matrix $A \in \mathbb{R}^{m \times n}$, the polar decomposition is defined as $A = UP$ where $U \in \mathbb{R}^{n \times m}$ and $...
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Why $L^{-1}U^T=D$ in LU decomposition?

I learned that given a matrix $A$, we can apply LU decomposition to get $A=LU$, where $L$ is lower triangular and $U$ is upper triangular. Further, if $A$ is symmetric (or Hermitian for complex $A$), ...
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Growth factor of specific matrix

Recalling that $$g_n(A) = \frac{||U||_{max}}{||A||_{max}}$$ where $A=LU$ with $U$ being upper triangular and $L$ unit lower triangular, and $|| \cdot ||_{max}$ is the largest element of $(|a_{ij}|)$. ...
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Define a matrix square root that preserve regularity

Let $A:\mathbb{R}\to \mathbb{R}^{n\times m}$ and $B\in \mathbb{R}^{n\times k}$. Is it possible to define $C:\mathbb{R}\to \mathbb{R}^{n\times m}$ satisfying the following two properties: for all $t\...
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Determine if LU decomposition is possible on a matrix?

I am trying to understand how you determine if LU decomposition is possible on a given matrix. I believe the way to calculate this is to check if the leading-matrices have non-zero determinants. I ...
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How to decompose matrix to its composition of rotation and scaling?

Ans: for rotation and scaling factor are $\frac{\pi}{6}$ and 8 respectively. I found a related question, but it wasn't explaining it well. I also understand scale and rotation separately but cannot ...
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Decomposition of orthogonal complex matrix

Note : $\mathrm{O}(n)$ is $\mathrm{O}(n, \mathbb{R})$. I tried to solve the following exercise : For every complex orthogonal matrix $g \in \mathrm{O}(n, \mathbb{C})$, i.e. $g^{-1} = {}^\mathrm{t}g$, ...
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A matrix as the product of other two matrices.

Assume $A$ be matrix of order $m×n$. Can we write the $A$ as product of two other matrices say, $P$ $(m×l)$ and $B$ $(l×n)$. That is $$A=PB$$ Is this possible for every matrices?
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A matrix decomposition problem similar to the CS decomposition

For two column orthonormal $X,Y\in\mathbb{C}^{n\times k},k<n$, prove that there exist unitary $Q\in\mathbb{C}^{n\times n},U,V\in\mathbb{C}^{k\times k}$, such that $QXU= \begin{bmatrix} I_k\\ 0 \end{...
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3 votes
3 answers
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Square root of a square matrix of ones

Let $J_n$ be the square matrix of ones (all entries are a one). I want to characterize the "square root" of $J_n$ where the square root is a matrix $A$ such that $A'A=J_n$ and $A$ is $k\...
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How to find eigenvectors of a matrix given only eigenvalues?

I'm trying to solve this problem: Given a $3\times 3$ matrix with eigen values $3,0,-1$, are their associated eigen vectors $(v_1,v_2,v_3)$ orthogonal to each other? My thought process is that we ...
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2 votes
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Is there a formula for the norm of an orthogonal projection?

In all introductory linear algebra texts there is a discussion on orthogonal projection. Let $u = w_1 + w_2$, where $w_1$ is the projection of $u$ along $v$ and $w_2$ is projection of orthogonal to $v$...
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Sufficient condition for $\langle Ax, Ay\rangle\leq \lambda^2_{max}\langle x, y\rangle$

Question. Is there a ''non-trivial'' sufficient condition on $x, y\in \mathbb R^n$ such that $$\langle Ax, Ay\rangle\leq \lambda^2_{\max}\langle x, y\rangle$$ where $A\in \mathbb R^{m\times n}$ and $\...
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Inverse of matrix using gauss Jordan method

When we use gauss Jordan elimination to find inverse of a matrix in the intermediate step we get a augmented matrix like [ U | ${L}^{-1}$] but for LU decomposition we use A = LU then why we get ${L}...
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How does the SVD behave under a "phase transformation"?

The singular value decomposition of a complex matrix $A$ can be written as $$ A= UDV^{H} $$ where $U$ and $V$ are hermitian matrices and $D$ is a diagonal matrix with entries $(D)_{ii}=\sigma_i\geq0$ ...
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Eigenvalues of a matrix whose elements are given as the sum of the product of another matrix's eigenvalues times some real coefficients

The title is quite long but I wouldn't know how to explain this otherwise. I have been working on this paper for quite some time, https://www.sciencedirect.com/.../pii/0393044088900319, and I can't ...
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$(w,h)=\arg\min_{x,y}\|A-xy\|^2_F,$ where $x,y$ are nonnegative vectors

Suppose $A\in \mathbb{R}^{m\times n},x\in\mathbb{R}_+^{m\times 1},y\in\mathbb{R}_+^{1\times n}$. $A$ has exactly one negative entry and others are nonnegative. Consider the problem $(w,h)=\...
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Determining a basis for the column space of A from its LU factorisation?

There are two ways of finding a basis for the column space of A from A = LU. The pivot columns in U correspond to the linearly independent columns of A, which in turn form a basis for the column ...
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Some questions of the proof related to Low-rank approximation on wikipedia

i lost the way when i read the proof of Eckart–Young–Mirsky theorem (for Frobenius norm) recorded on wikipedia(enter link description here). i firstly want to know in the following logitic of the ...
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Approximating inverse square root of diagonal matrix multiplied with orthonormal vectors

For $d>m,$ let $Q \in \mathbb{R}^{d \times m}$ with $Q^{T} Q = I_m$ and $D \in \mathbb{R}^{d \times d}$ be diagonal matrix with positive elements. I wonder there exists any results about $Q^T D^{-1/...
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Do the rows used in row operations during LU factorisation matter?

A method I have seen for finding the LU factorisation of a matrix is that U is the row echelon form of A. The row operations we perform on A to get to U must involve replacing $R_i$ by $R_i - kR_j$ ...
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Householder QR with column pivoting

In numerical linear algebra, we know how Householder QR can handle rank deficient matrices with column pivoting: which is essentially to choose the left-over columns with the maximum norm and use ...
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Numerical low-rank approximation of sample covariance matrix

I could use some help with a matrix optimization problem. I have a $M \times M$ sample covariance matrix $\hat{\mathbf{C}}$ that I'd like to approximate using the decomposition $\hat{\mathbf{C}} = \...
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