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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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in Algorithms for Non-negative Matrix Factorization, the proofs of convergence, why is G(h,h') defined like this? [on hold]

because the auxiliary function has two different forms in this paper. h' has any special meaning? paper: Algorithms for Non-negative Matrix Factorization section: 6 Proofs of convergence Definition 1
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387 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 ...
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pivoted QR decomposition - weighted least-norm

Using the QR decomposition for the underdetermined weighted least-norm problem: min$||W^{-1}x||^2$ st. $Ax=b$, $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$ $A \in \mathbb{R}^{m \times n}$, $...
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1answer
5k views

Orthogonal decomposition of vector v with respect to span W

I need to find the orthogonal decomposition of vector V with respect to span W $$ v= \begin{bmatrix} 2 \\ -1 \\ 5 \\ 6 \\ \end{bmatrix} $$ $$ w = span \begin{pmatrix} ...
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Question of the Cholesky decomposition of symmetric positive definite matrix

This is a exercise on my numerical analysis textbook: Suppose $\mathbf A$ is a positive-definite symmetric matrix, and the Cholesky decomposition is of the form $\mathbf {A} =\mathbf {LL}^{T}$, ...
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Matrix decomposition into 2x2 elementary transforms

Rotations matrices can be decomposed into a product of n(n-1)/2 elementary rotations operating on only two coordinates. Similarly, can any square matrix be decomposed into a product of n(n-1)/2 ...
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1answer
119 views

Show that $\mathbf{X}_{tr} \leq \sqrt{nm} f(\mathbf{X})$

$\mathbf{X}$ is a $n\times m$ matrix and $$ f(\mathbf{X}) = \min\limits_{X = UV} \max_i \mathbf{\|U_i\|} \max_j \mathbf{\|V_j\|} \;\; \text{(max over the rows) and } \|\| \text{is the $l_2$ norm} $$ ...
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Rank-1 modification of correlation matrix

I inherited a problem at work and have trouble wrapping my head around it. It doesn't help I'm insufficiently sophisticated with linalg. Help with or pointing to appropriate literature will be much ...
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16 views

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
678 views

Householder reduction to Hessenberg form

I've read somewhere that Hessenberg decomposition is not unique unless the first column of $Q$ in $Q^{T}AQ =H$ is specified. But then, if I am given a matrix $A \in R^{n \times n}$, I can apply the ...
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An extra row on Smith Normal Form

Let's suppose I have an integer matrix $M$ of size $m\times n$ and I know its Smith Normal Form $S$. Can I say something of a matrix $M'$ of size $(m+1)\times n$ which consists of $M$ with an extra ...
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1answer
23 views

Does PCA always have to reduce dimensionality?

I came across this paper where the authors implement a regularized learning model to estimate the covariance matrix of a dataset. The authors say they "...propose a regularized form of Principal ...
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37 views
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33 views

Least-Squares Opposite

Is there an opposite formulation of least-squares projection where the distance between each point–say, $(x_{i}, y_{i})$– and the subspace that it's projected onto (e.g. a line through the origin) is ...
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75 views

Vandermonde decomposition of Toeplitz matrix

Since every Hermitian Toeplitz $A \in C^{n \times n}$ with $rank(A)=K$ can be decomposed as $$ A = VDV^T $$ where $V \in C^{n \times K}$ is a Vandermonde matrix, and $D \in R^{K \times K}$ is a ...
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LU Decomposition vs. Cholesky Decomposition

What is the difference between LU Decomposition and Cholesky Decomposition about using these methods to solving linear equation systems? Could you explain the difference with a simple example? Also ...
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1answer
73 views

Prove that if $M$ is a symetric positive definite matrix then $(S^T)MS$ is also symetric positive definite [closed]

I'm asked to prove that with $S$ being any non singular matrix , if $M$ is a symetric positive definite matrix then $S^TMS$ is also symetric positive definite.
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26 views

Solve Ax=b using Cholesky decomposition

I was reading the following article about the direct stiffness method. When it comes to solving the system of equations: The site states: [...]There are several different methods available for ...
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1answer
33 views

Positivity of a matrix

Let $A$ be a $3\times 3$ matrix defined in the following way: $$A=\begin{bmatrix} a & c & 0\\c & b &-c\\0 & -c & 1-a-b\end{bmatrix}$$ I wish to show that $A=BB^t$ for some ...
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1answer
44 views

Converting Jordan Normal Form into Real Jordan Form

Given the matrix $$\begin{bmatrix} 0 & 0 & 0 & -8\\ 1 & 0 & 0& 16 \\ 0 & 1 & 0 & -14 \\ 0 & 0 & 1 & 6 \\ \end{bmatrix}$$ ...
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1answer
44 views

Eigendecomposition of $(a^\top X a)X - Xaa^\top X$

Let $a\in\mathbb{R}^n$ be a nonzero vector, $X\in\mathbb{R}^{n\times n}$ be positive definite. What are the eigenvalues and eigenvectors of $(a^\top X a) X - Xaa^\top X$?
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1answer
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QR factorization: Show column spaces of $A$ and $Q$ are equal.

If I have a $A=QR$ factorization by doing the Gram-Schmidt process, how can I prove that the column space of $A$ and the column space of $Q$ are same?
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1answer
690 views

Is it possible to compute derivative of truncated SVD without computing a full SVD?

I am working with derivatives of the SVD. My setting is the following: Let $A:(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}^{m\times n}$, $t\mapsto A(t)$ be a differentiable matrix-valued function, ...
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Relationships between the singular values and eigenvalues of an asymmetric matrix A

As we know, if A is a real and symmetric matrix, the SVD decomposition is just the eigendecomposition, and the singular values and singular vectors are just eigenvalues and eigenvectors. For the case ...
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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
44 views

Solving a system by using Cholesky Decomposition $(LDL^T)$

$$\begin{bmatrix} 4 & 1 & 1 \\ 1 & 2 & -1 \\ 1 & -1 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ ...
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22 views

Applications where weighted singular values are useful

Let $A = U\Sigma V^*$ by the compact SVD where rank$(A)=r$ and $\Sigma$ is $r\times r$. If $A$ is Hermitian, then $U=V$. Let us form another matrix $A_k = UK\Sigma V^*$, where $K\ne I$ is positive ...
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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 ...
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2answers
109 views

Questions on PCA

I have a hard time understand the following statements below about PCA (normed or not normed). a) the matrix to diagonalize is the matrix of linear correlations of original variables. b) An ...
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1answer
34 views

A = B * C, how do I solve for B

$$A = B C$$ where $A$'s dimension = $n$ x $1$, $B$ 's dimension = $n$ x $n$, C 's dimension = $A$ = $n$ x $1$ I know A and C. How do I solve for B? Attempt: I was thinking about multiply by the ...
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Intuition why eigen decomposition is equivalent to PCA

I understand that the idea of principle component analysis is to find the projection onto a vector with the largest variance. The book says this can be achieved with eigen decomposition of the ...
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Prove $A>P^{T}B^{-1}P$ iff $B>PA^{-1}P^{T}$.

Suppose $A,B$ are $n\times n$ positive definite real symmetric matrices,P is an $n\times n$ real matrix, prove that $A>P^{T}B^{-1}P$ iff $B>PA^{-1}P^{T}$. By using the orthogonal ...
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29 views

Given two side matrices $P$ and $Q$, extract (find) the diagonal scaling matrix $\Sigma$ of a singular value decomposition

I have an application where I have already approximated a given matrix $R$ of size $m \times n$ by multiplying two matrices $P$ and $Q^{\mathrm T}$: $\hat R=PQ^{\mathrm T}$. $P$ is size $m \times k$ ...
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27 views

Inverse of a triangle matrix

I have been given an assignment where I need to find the inverse of an upper triangular matrix. I have been given these two formulas and am to use Lemma $2$. Can someone explain to me how this works. ...
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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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63 views

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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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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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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1answer
49 views

Iwasawa decomposition of inverse

Let $G$ be a semisimple rank one Lie group with finite center. Let $G=KAN$ be the Iwasawa decomposition with $\mathfrak{a}=$Lie($A)=\text{span}\ H$. Then if $G\ni g=kan, a=exp(tH)$ is it true that $$...
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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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1answer
717 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom

There are a couple of problems and solutions where affine matrices are decomposed into their separate transformations. However, they are all for the 2D case and I`m finding it difficult to generalise ...
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1answer
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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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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
54 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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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
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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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Block-diagonalizing an antisymmetric matrix

I was wondering how to block-diagonalize a $10 \times 10$ antisymmetric matrix into block matrices along the diagonal. Can I just diagonalize each non-diagonal block? Thanks!
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2answers
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What do you call this equivalence relation? $A \simeq B$ if $A = P^t BP$ for some invertible matrix $P$

If $A, B$ are square matrices with coefficients in some ring, we say that $A$ is similar to $B$ if $A = PBP^{-1}$ for some invertible matrix $P$. Similar matrices represent the same linear operator ...
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1answer
24 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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245 views

eigenvectors of a circulant block matrix

I am looking to find eigenvectors of circulant block matrices. I have a matrix given by: $$ M= \begin{pmatrix}Z& A\\ B & Z \end{pmatrix} $$ where $Z$ is an $n\times n$ zero matrix, ...