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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What are the meanings of matrix constructs of pre-multiplication of the matrix transpose and post-multiplication by the matrix itself.

In many books and articles I sometimes see similar matrix constructs, multiplication of the matrix transpose (or inverse), some other matrix, and then the first matrix itself, like this: $M^T AM$ or $...
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Solve many linear equations of similar structure

Given G: real and symmetric square matrix v: real column vector I need to solve n linear systems of the form \begin{align} A = \begin{pmatrix} G & v \\\ v^T & 0 \end{pmatrix}\end{align} \...
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Is posible represent a matrix $A$ like $A=\lambda U$ where $U$ is a unitary matrix and $\lambda \in \mathbb{C}$?

Determine if is possible to represent the matriz $$\begin{equation} \begin{pmatrix} i & 0 & 2\\ 2i & 0 & -1\\ 0 & -(1+2i) & 0 \end{pmatrix} \end{equation}$$ like $A=\lambda U$ ...
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Schur Triangularization

Let $A$ and $B$ be two square matrices such that $AB=BA$. Schur decomposition of both of $A$ and $B$ is $A=UDU^{*}$, $B=UTU^{*}$ i.e they have the same unitary matrix $U$ and if $A$ and $B$ are ...
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Convergence of the complex QR algorithm to Schur decomposition

I study the complex Schur decomposition of a complex matrix $A \in \mathbb{C}^{n \times n}$, that is: $$ A = U T U^H $$ where $T$ is upper-triangular (the eigenvalues of $A$ appear on its diagonal, ...
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Compute matrix product and inverse expression using LU-decomposition

Given Matrices $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{m \times n}$. I want to compute $BA^{-1}$ without directly computing the inverse of $A$. However I can use: Matrix multiplication ...
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Would symmetry positive semi-definite matrix always decomposable?

Given symmetry positive semi-definite matrix $A \in R^{n\times n}$. And $Det(A) \geq 0$. Would there always exist real matrix $B$, such that $A = B \cdot B^T$? If so why? Or why not?
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QR algorithm with reduced Hessenberg form

Let $H$ be an upper Hessenberg matrix (see Wikipedia). I want to compute its complex Schur decomposition $H = U T U^H$ where $U^H = U^{-1}$ is unitary and $T$ is upper triangular matrix (the diagonal ...
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Solving a linear matrix equation with both left and right multiplication of unknown

I would like to solve a matrix equation of the form $$ \mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{A}^T = \mathbf{B} $$ where $\mathbf{A}$ and $\mathbf{B}$ are known $n \times n$ matrices, and $\...
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Solving for $X$ using the SVD of $QX$ when $Q$ is orthogonal

I inherited some code (see below), and I am not quite sure what it does. It is part of a factor analysis-type model that learns a latent variable $X \in \mathbb{R}^{N \times K}$ with $N > K$ that ...
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Why does SVD solve $\underset{U,V}{\min}\| A - UV^T\|_F^2$

I read here the following: You can solve the quadratic problem below through Singular Value Decomposition (SVD) of the matrix $A$. \begin{align} \underset{U,V}{\min} \| A - UV^T\|_F^2 \end{...
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Householder algorithm for QR-decomposition: how to retrieve Q?

Suppose that $A$ is a non-singular real $n\times n$-matrix. Consider $Ax=b$ and $A=QR$. One can solve the equation by using following algorithm, with input $n$ and $C:=(A\mid b)\in\mathbb{R}^{(n+1)\...
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Bartels-Stewart Algorithm for the Complex case

Let $$ A X + X B = C $$ be the Sylvester equation when $A,B,C \in \mathbb{C}^{n \times n}$ are complex matrices. I want to solve it for $X$. Python's SciPy package $\texttt{solve_sylvester}$ function ...
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Solving a matrix equation given an eigenvector

If $v(z=0)$ is chosen to be an eigenvector of $A$ describe the solution evaluated for $v(nδ)$, for any integer $n$ for the following: $$(NI-A)v(z+δ)=(NI+A)v(z)$$ where $N$ is an arbitrary number. ...
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If W1= H1' and W2 = H2', and Vopt = [H1 | H2]'. Is there any V in terms of W1 and W2 that is equal to Vopt?

Let $H_{1} \in \mathbb{C}^{4\times 2}$ and $H_{2} \in \mathbb{C}^{4\times 2}$ be i.i.d. zero-mean complex Gaussian random variables. We design $W_{1}$ and $W_{2}$ as $W_{1} = H_{1}^{\dagger}$, and $W_{...
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Solving $max \sum_{i=1}^2 \sum_{j=1}^2 (\phi_i A_{ij} \psi_j)^2$

Could you please provide some hints for me to solve this optimization problem? Here, for any $i=1,2$ and $j=1,2$, $\phi_i$ and $\psi_j$ are unknown vectors, $\alpha_i$, $\beta_j$ are some known ...
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Eigenvectors of Matrix A [closed]

For a matrix Q prove the eigenvectors of kI-Q are equal to that of Q
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Take derivative of matrix

A part of an objective function is: $$F=\|H-\mu_H\|_F^2$$ And we have: $$\mu_H=\frac{\Sigma H}{n_H}$$ In fact, $\mu_H$ is the average of $H$ in one dimension and is repeated $n$ times in which all ...
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Estimating matrix raised to a power

What would be the best and easiest way to estimate a matrix to a power? Specifically, let's say you have matrix A: $$A = \left[ \begin{matrix} 3/4 & 1/4 & 0 & 0 \\\ 1/4 ...
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What are the most relevant applications of polar decomposition?

Assume there exists a new and very efficient algorithm for calculating the polar decomposition of a matrix $A=UP$, where $U$ is a unitary matrix and $P$ is a positive-semidefinite Hermitian matrix. ...
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Diagonal Elements of Inverse of Symmetric Matrix.

Is there any shortcuts to calculate diagonal elements of the inverse of a symmetric matrix: $$diag(X^T\cdot X)^{-1} = ?$$ Probably X is a sparse matrix. ($X^{T}X$ - symmetric matrix)
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Do Jordan chains single out special directions of an eigenspace?

Let $T$ be a linear operator on $\mathbb C^n$ where $n<\infty$ and let $U$ be the eigenspace associated with eigenvalue $\lambda$. For simplicity, assume $\lambda$ is the only eigenvalue of $T$. ...
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Proof that transposed matrix can be used for ZCA

I am using Zero-phase Component Analysis (ZCA) for the processing of images. The images are provided as a matrix $X$ with $m=$ number of rows = number of images and $n=$ number of features (pixels) = ...
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Find all solutions of least squares problem

I have the following exercise (this is exercise 4.39 of Fundamentals of Matrix Comuptations - Watkins) : I am not sure about how to find all the solutions(item e). I think I must use itens c) and d) ...
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42 views

Derivative of $P^{-1}HP$ w.r.t. $H$

Is there a matrix expression for $\frac{d}{dH}P^{-1}HP$ (for constant square matrix $P$)? Background (if necessary) I seek to design an objective function (and its gradient) over the 8D space of ...
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Find SVD of $A (A^T A)^{-1} A^T$

This is exercise 4.2.12 of Fundamentals of matrix Computations from Watkins: Let $A \in\mathbb R^{n\times m}$, $n \geq m$, $\operatorname{rank}(A) = m$ with complete SVD $ A = U\Sigma V^T$, $U\in \...
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Need help to explain the objective function with numerical example.

ai = element in i position in matrix? Cj= column j? Xc(i,j) = X in cluster following (i,j) position in matrix?
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How to find proper SVD components?

My approach is based on this method: we want to find $U$, $\Sigma$, $V$ so that $A = U \Sigma V$. Then $A^T A = V^T \Sigma U^T U \Sigma V$. Since $U$ is an orthogonal matrix, this equals $V^T \Sigma^...
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Fundamentals of Matrix Computations, Watkins, exercise $4.3.9(e)$, SVD.

Given that $$A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6\end{bmatrix}, \qquad b=\begin{bmatrix} 1 \\ 1\\ 1\end{bmatrix},$$ what is the method to find all solutions of the least-squares ...
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Matrix inverse with diagonal Perturbations

Consider $$ P:= (X^TX+D_1^{-1})^{-1} - (X^TX+D_2^{-1})^{-1} $$ $$ Q:= (X^TX+D_1^{-1})^{-1/2} - (X^TX+D_2^{-1})^{-1/2} $$ $X$ is an $n \times p$ matrix with $p > n$ and $D_1, D_2$ are diagonal ...
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Are elementary row operators in linear algebra mutually exclusive?

There are three types of elementary row operations: I) row switching, II) row multiplication and III) row addition, corresponding to three kinds of row operation matrix. My question is that does ...
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Two rectangular matrices are equivalent matrices if and only if they have the same rank

We will call two rectangular matrices $A$ and $B$ of the same dimensions equivalent if there exist two non-singular matrices $P$ and $Q$ such that $B = PAQ$ prove that $A$ and $B$ are equivalent if ...
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How to solve the $Var(AX^1)$?

Let the partition X is: $X=\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=\begin{bmatrix}X^1\\---\\X^2\end{bmatrix}$, $μ_x=\begin{bmatrix}4\\3\\2\\1\end{bmatrix}$ Variance-Covariance matrix: $Σ_x=...
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Property of the projection onto Singular Vectors

I come across this property while reading a paper and cannot find it anywhere: The property goes as follow: given a matrix $A \in R^{N\times n}$, denote $ U,V $ as the subspaces spanned by left and ...
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Is a symmetric matrix positive definite iff $D$ in its LDU decomposition is positive definite?

Given $$A=LDU$$ where $A$ is a real symmetric matrix $L$ is a lower unitriangular matrix $D$ is a diagonal matrix $U$ is an upper unitriangular matrix can we say that $$A>0 \iff D>0$$ ? Edit:...
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Linear Transformation of polynomial with ordered bases

I have tried to solve the following:- Let $P_2 -> P_2$ be defined by $$T(a+bx+cx^2) = 2b+3cx+(a-b)x^2$$ Find the matrix of transformation w.r.t ordered bases $B_1= \left\{x^2, x^2+x, x^2+x+1\right\}...
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Some problems with Exponential matrix

We've know that $e^A$ is Orthogonal when $A$ is Real anti-symmetric, how can we know that for any orthogonal matrix we can find the real anti-symmetric matrix. Similarly, for one unitary matrix $U$, ...
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Given A=LU factorization, prove that the basis of column space A is the columns of L that correspond to the pivot columns of U

I understand that the basis of column space A is just the columns of A that correspond to the pivot columns of U. This is because U is just the reduced row echelon form. However, as mentioned in the ...
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I'm now stuck in how to convert the distance matrix to the real coordinates of points by using $M_{ij} = \frac {D^2_{1j}+D^2_{i1}-D^2_{ij}} 2 \,.$

I want to implement $M_{ij} = \frac {D^2_{1j}+D^2_{i1}-D^2_{ij}} 2 \,$to find the coordinates of points from distance matrix. And we already know one point is original point (0,0) which can be regard ...
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Show that $A$ is positive definite via the Cholesky decomposition

I have calculated the Cholesky decomposition of the matrix \begin{equation*}A=\begin{pmatrix}1 & -1 & -1 & 0 \\ -1 & 5 & 5 & -4 \\ -1 & 5 & 6 & -3 \\ 0 & -4 &...
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Relationship between segre and weyr characteristic of a matrix

The problem is in the problem of Roger A.Horn's textbook Matrix analysis,3rd.The main puzzle of me is how to estabilish the relationship between $S_{w_k}$ and $W_{S_k}$. Can you give me some hints to ...
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Idempotent matrix of the form $(D-A)$

Does there exist an idempotent matrix of the form $P=(D-A)$ where $P^2 = P$ if $A$ is idempotent? $D$ is a diagonal matrix with positive distinct entries. For the trivial case when $D$ is the identity ...
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Norm of inverse of all one matrix plus a PSD matrix

Consider the operator norm of the following $n \times n$ matrix: $$ \|(I + 11^* + X^*X)^{-1} 11^* (I + 11^* + X^*X)^{-1}\|, $$ where X is a $n \times n$ matrix. Is it bounded by $\frac{1}{n}$? I ...
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Finding the inverse of the eigenvectors matrix

Hi I have the following $n\times n $ matrix: $$\left(\begin{array}{cccccc} 0 & 0 & 0 & 0 & & -0.5\\ 0.5 & 0 & 0 & 0 & & -0.5\\ 0 & 0.5 & 0 & 0 &...
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Square root of the product of a diagonal and symmetric matrix

If I have a diagonal matrix $D$ and a positive-definite symmetric matrix $C$, is there a formula for the square root of the product, $(DC)^{1/2}$? Also, $DC \neq CD$. What I have so far is \begin{...
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Determinant of Square Root of Positive Define Matrix

Suppose the matrix $A \in \mathbb{R}^{n\times n}$ is positive definite symmetric. To begin, I want to investigate if the following equality holds $$ |\det A^{1/2}| = |\det A|^{1/2}. $$ Since $A$ is ...
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If $A$ is symmetric and PD then $B$ is symmetric

Let $A$ be a symmetric and PD matrix. Prove that the matrix $B=A_2-\ell_1u_1^T$ which you get in the LU decomposition process is symmetric. I understand that $A_2$ is the sub-matrix of $A$ (from $(2,2)...
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Finding a matrix that can be represented with only single LU decomposition

I'm trying to disprove the following statement: Let $M$ be a singular matrix $3\times 3$ that can be represented with LU decomposition ($M=LU$), then the decomposition is unique (only one ...
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27 views

Extra parameters necessary to reconstruct a matrix (up to a rotation), given its eigenvalues

We are given 3 real eigenvalues $\lambda_1 > \lambda_2 > \lambda_3$ of a $3 \times 3$ real symmetric matrix $A$. Which are the extra real parameters that are needed in order to reconstruct $A$ ...
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What does Canonical Form Means Intuitively?

I have seen multiple times where canonical form is mentioned. I went to Wikipedia and as usual its quite formal definition and not intuitive at all. So the following context taken from MathWorld, what ...

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