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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?

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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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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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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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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How to make sure matrix completion can generate a matrix with values in expected range?

I am doing a matrix completion project. Assume that I have an incomplete matrix like ...
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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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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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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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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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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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30 views

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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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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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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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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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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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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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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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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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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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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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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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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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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24 views

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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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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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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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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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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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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50 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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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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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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50 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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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 ...