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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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Using Cholesky decomposition to compute covariance matrix determinant

I am reading through this paper to try and code the model myself. The specifics of the paper don't matter, however in the authors matlab code I noticed they use a Cholesky decomposition instead of ...
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Rewrite a condition on a $3\times3$ matrix

Consider the $3\times 3$ matrix $$ A\equiv \begin{pmatrix} \mu_1-\mu_1'& \mu_1-\mu_1'-c & \mu_1-\mu_1'-c-d\\ \mu_1+a-\mu_1'& \mu_1+a-\mu_1'-c & \mu_1+a-\mu_1'-c-d\\ \mu_1+a+b-\mu_1'&...
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How to obtain the Q matrix from QR decomposition using Fast Givens rotation [on hold]

I am not able to obtain Q matrix correctly when I attempt QR decomposition using fast Givens rotation. Would somebody kindly help along with a numerical example?
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1answer
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A convex combination of unitary transforms converts any matrix to identity

Question Show that there exists a set of unitary matrices $\{U_i\}$, and probability $\{p_i\}$, such that for any $n \times n$ matrix $A$ \begin{equation} \tag{1} \sum_{i} p_i U_i A U^{\dagger}_i = \...
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1answer
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How many possible factorizations are there for a square matrix, and how can we know?

Given a square matrix A, how many possible factorization CB=A is there, and how can this number be calculated? I understand that there are many ways of decomposing a matrix that yields matrix ...
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19 views

Confusion with the QR decomposition

I have some trouble to understand this: According to the factorization QR, Given a matrix $A\in\mathbb{R}^{n\times p}$, with $rank=min(n,p)$, there exists an orthogonal matrix $Q\in\mathbb{R}^{n\...
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32 views
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Smith normal form of rectangular matrix in MATLAB

Suppose I've got a nonsquare integer matrix, say $\begin{pmatrix}3 & 1 & 1 & 1\\1 & 1 & 1 &1\end{pmatrix}$ and want to compute its Smith normal form--in this case, $\begin{...
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42 views

complexity of QR decomposition .

Could anyone help me to find the complexity of QR decomposition of matrix $A \in C^{N \times P}$, where $P \leq N$. I am also willing to know the complexity of adding a matrix $A \in C^{N \times N}$ ...
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1answer
35 views

Polar Decomposition of 2x2 Matrix

I have the following homework problem and I just don't know how to go about starting it. Is it asking me to find a unique value of ϕ? I just can't see any other solution apart from when ϕ = θ. So my ...
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20 views

Understanding singular value decomposition example

I wanted to view SVD in action (using Octave) by running it on an image and then breaking it down into a set of rank 1 matrices. I'm getting stuck before that though, because I'm unable to reproduce ...
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1answer
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Linear Transform T = A + B. When is Tw = Aw + Bw such that Tw is T restricted to the domain W

I have a question related to the Jordan-Chevalley Decomposition but I am also wondering about the general case. I have that V is a finite dimensional vector space over $\mathbb{C}$ and $T:V\...
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2answers
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Why is the 'controllable subspace' actually controllable?

I am looking at the Kalman decomposition of a linear system into 'controllable' and 'uncontrollble' subspaces. The references I am using are these lecture notes and section 3.3 of 'Robust and Optimal ...
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1answer
22 views

Representation of a matrix (tensor)

Let us consider the following $2 \times 2$ matrix, $A$. $$ A = \begin{bmatrix} w_1^TP_{11}w_1 & w_1^TP_{12}w_2 \\ w_2^TP_{21}w_1 & w_2^TP_{22}w_2 \end{bmatrix} $$ where $P_{ij}$'s are $n\...
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1answer
51 views

Solving an undetermined or overdetermined system of equations with constraints

I have a table that looks like this: I would like to determine the values for each of the different categories in the columns, such that col1*col2*col3 equal what'...
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Cholesky factor when adding a row and column in between

I have a problem where I have the Cholesky factorization ($A=LL'$) of a symmetric positive-definite matrix. Now, I need to add a new row and column somewhere in the "middle" of the matrix and compute ...
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Choleskey Decomposition of a Complex Matrix in MATLAB

I am trying to decompose a cross spectral density matrix (A Complex Matrix) using "chol" command in MATLAB. we know that every positive definite and Hermitian matrix can be decomposed using Cholesky ...
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1answer
50 views

Factoring a matrix as the product of block triangular and diagonal matrices.

How can I check that the matrix $$K = \left[\begin{array}{c|cc} 1 & 0^{\mathrm T}_m & m \mathbf u^{\mathrm T} \\ \hline 0_m & I_m & I_m \\ m \mathbf u & I_m & O_m \end{array}...
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Positive Definiteness of sub-matrices when performing Gaussian elimination. [duplicate]

Let $A$ be a symmetric positive definite matrix. Is there a simple way to show, perhaps using a well known result, that the submatricies obtained by applying Gaussian elimination, $A^{(k)}= (a^{(k)}_{...
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1answer
39 views

Inverting a matrix with the same diagonal entries in a particular form

Hi I'm struggling with this inversion and any help would be greatly appreciated. I want to invert the following $\mathbb R^{m\times m}$ matrix \begin{bmatrix} 1 + m & m & \dots &...
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1answer
38 views

Show that $A$ is normal if it commutes with some normal matrix with distinct eigenvalues.

Suppose that $\mathbf{A} \in \mathbb{C}^{n \times n}$ and there is a normal matrix $\mathbf{X} \in \mathbb{C}^{n \times n}$ such that $\mathbf{X}$ has distinct eigenvalues (none of them repeat) and $\...
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What is the relevance of this theorem? (Complete orthogonal decomposition)

I'm reading the book Matrix Analysis for Scientists and Engineers by Alan J. Laub. In the chapter about Canonical forms, the following theorem is presented: Theorem 10.25 (Complete Orthogonal ...
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Convergent series description of ratio of two bilinear forms

I have to numerically calculate the ratio of two bilinear forms: $\frac{x_1}{x_2} = \frac{v^T U_1 v}{v^T U_2 v}$, where $U_1$ and $U_2$ are unitary matrices. Both bilinear forms $x_1$ and $x_2$ are ...
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Creating a correlated distribution with an indefinite matrix

For the past week, I've been trying to figure out how to implement a linear transformation of a random variable vector with entries $h_{i}$ of the form $\tilde h = Wh$ where h is a non correlated ...
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Which type of factorization is appropriate?

I have a symmetric matrix (square)($A$) with positive values and zero on its main diagonal. I need to find a matrix $Y$ which is: $Y^TY = A$ I don't have any non-negativity constraint on the ...
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How to get the partial information matrix from the covariance matrix

I have four random variables${\left( {x,y,z,\varphi} \right)^T}$. And I know its covariance matrix $Cov=\left[ {\begin{array}{*{20}{c}} {\sigma _{xx}^2}&{\sigma _{yx}^2}&{\sigma _{zx}^2}&{...
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Approximation of special differential operator applied to special function on $\operatorname{SL}_2(\mathbb{R})$

Let $G = \operatorname{SL}_2 (\mathbb{R})$ and $G=ANK$ be the Iwasawa decomposition, so for $g \in G$ we write $g = a\hat{n}k$ where $$a=\begin{pmatrix}\hat{a} & \\ & \hat{a}^{-1}\end{pmatrix}\...
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Can we do Karhunen-Loeve Transformation in one dimension matrix?

I have a one-dimensional matrix (that was extracted from the audio data) to process so the noise of the data could be eliminated. A paper told me that I have to use the Karhunen-Loeve Transformation (...
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28 views

Minimizing $||A - (M^2) (N^2)||_F^2$ vs NMF

A reasonable way to factor a non-negative matrix $A$ into two non-negative matrices $M$ and $N$ can be done by minimizing the squared Frobenius norm, $||A - (M^2) (N^2)||_F^2$, where $M^2$ is the ...
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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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49 views

Do Givens matrices commute?

Suppose I have Givens matrices (as defined here), $G_1,...G_k$, and some permutation $P$ which mapping $\{1,..., k\}$ to $\{1,..., k\}$. Is it true that: $$G_1G_2... G_k = G_{p(1)}G_{p(2)}...G_{p(k)}$...
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What is the most general matrix that gives rise to all even characteristic polynomials?

Is there some general form of all matrices which give rise to all even or all odd characteristic polynomial terms? For skew-symmetric matrices such that $A^T=-A$ we necessarily have all even or all ...
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1answer
17 views

Multiply diagonal of matrix by a set of values

Is there any unitary transformation that has the effect of only multiplying the diagonal with some values. For example if I start with the matrix $$A=\left(\matrix{1&a&b&c\\d&1&e&...
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Householder matrices for thin QR updating

Suppose you have a QR decomposition of the form: $X=QR$ (Where $X$ is an arbitrary matrix of size $(n \times p)$, $Q$ is an orthogonal matrix of size $(n \times n)$ and $R$ is an upper trapezoidal ...
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1answer
92 views

What type of matrices lead to $\pm$ eigenvalues?

I have a code which is spitting out matrices of the form $$\left(\matrix{0&a&-a\\a&+\gamma&0\\-a&0&-\gamma}\right)$$ It has trace $0$ and thus its eigenvalues are of the form ...
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1answer
22 views

A rotation which switches two blocks of a matrix

I have a $6\times6$ matrix $A$ which I would like to transform using some unitary operator such that $B=U^\dagger AU$. I would like to swap the elements of two sub-blocks of the matrix. If the ...
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How to induct the Nonnegative Matrix Factorization on Orthogonal Subspace

I am studying the paper Nonnegative Matrix Factorization on Orthogonal Subspace I am sorry my reputation is too low to post the pic. The objective function is F and rewritten as J, I could not ...
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30 views

Rank of the element-wise square of a matrix

I am looking for a real matrix A of rank r with non-negative entries with the following property : For every complex matrix B such that $B\circ \overline B=A$, the rank of B is strictly greater than ...
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Transition matrix of a specific matrix

Let $ A = \begin{bmatrix} 3&3&3 \\ 3&3&3\\ 3&3&3\\ \end{bmatrix} $ I know that its rational canonical form is $ \begin{bmatrix} 0&0&0 \\ 0&0&0\\ 0&1&...
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Does Eigenvalue decomposition (EVD) of a matrix $\mathbf{A}$ mean $\mathbf{Q}$ in wiki,if $\mathbf{A}$ ignore the rank-1 constraint

In this paper ( https://liu.diva-portal.org/smash/get/diva2:1245887/FULLTEXT01.pdf ) the author told me two things 1.$\mathbf F_k= \mathbf f_k \mathbf f^H_k, \mathbf f_k$ is a $N$ by $1$ matrix,and $\...
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What is the relationship between $||.||_{max}$ and energy of a matrix?

I was interested to find a relationship between $||.||_{max}$, i.e. max-norm of a matrix with that of its energy. $||.||_{max}$ of a matrix is defined by the maximum entry in the matrix. Generally, ...
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Matrix Numerov Method in three dimensions

Hi can anybody help me to write the following equation in form of matrix by using Numerov's method. $\left(\frac{d^2}{d x_1^2}+\frac{d^2}{d x_2^2}+\frac{d^2}{d x_3^2}+x^2_1+x^2_2+x^2_3+x_1x_2+x_2x_3+...
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22 views

Have a matrix dot its transpose, what is the original matrix?

I have a matrix that equals the dot product of a matrix A with its transpose. How do I get the matrix A? Ex.: $AA^T$ = [a given matrix]
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Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
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circulant matrix inversion using fast fourier transform

I am Huda. May I know, if I have a circulant matrix, can I calculate its inversion using fast fourier transform? If yes, I really need an explanation. Thank you.
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Why is this algorithm for PLS correct?

I am studying Partial Least Square Regression (PLS), and I am not able to understand how the algorithm for performing a PLS factorization works. PLS is composed of two parts. First, we factor two ...
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1answer
36 views

Result for $A^{-\frac{1}{2}}A(A^{-\frac{1}{2}})^T$

Let $A_n$ be a symmetric square positive definite matrix. (A variance and covariance matrix.) So, can I say $A^{-\frac{1}{2}}A(A^{-\frac{1}{2}})^T = I_n$?
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LU decomposition of banded matrix without pivot

Say I have a Matrix $M$ that is a banded matrix where an LU decomposition exists (so without pivot). Are the $L$ and $U$ of the $LU$ decomposition also banded? Is the band the same (ignoring the upper/...
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2answers
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How to solve $Q$ matrix from Householder QR-factorization? - Lapack

I'm using the subroutine sgeqr2 from Lapack. This subroutine solves the QR-factorization $$A = QR$$ It's easy to find the $R$ matrix, because the in-out argument $A$ of subroutine sgeqr2 will return ...
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29 views

LU decomposition of matrix using column pivoting

I want to find the LU decomposition of the following matrix $A$ using Gauss algorithm and column pivoting. $$A=\begin{pmatrix}6 & 4 & 3 & 1\\ 1 & 1 & 0 & 2 \\ 2 & 3 & 1 ...
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Is there any way to use matrix decomposition for finding $A^n$?

If I want to take the power of matrix $A$ with e.g 3, $A^3$ or with power of $-\frac {1}{2}$, e.g $A^{-\frac {1}{2}}$ etc. Is there an easy way to solve $A^n$, where $n\in R$ and $A \in R^{nxn}$ by ...