# Questions tagged [matrix-decomposition]

Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.

595 questions
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### Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
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### Can I go from the LU factorization of a symmetric matrix to its Cholesky factorization, without starting over?

I mistakenly computed the LU factorization and then realized that the question is asking for a Cholesky factorization, i.e., finding a lower triangular matrix L such that the symmetric matrix A has ...
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### Fast arbitrary decomposition of a positive-definite matrix

Given a positive-definite $n\times n$ matrix $\mathbf{A}$, my goal is to present it as a product of the form $\mathbf{H^TH}$, where $\mathbf{H}$ is an arbitrary $n\times n$ matrix. Cholesky ...
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### Sufficient/necessary condition for submatrix determinant (minor) that decreases with size?

Definition. Given a square matrix ${\bf{A}}=[a_{ij}] \in {\mathbb{C}^{n \times n}}$, the submatrix ${{\bf{A}}_{{i_1},{i_2},...{i_k}}}$ is formed by retaining the $({i_1},{i_2},...{i_k})$-th rows and ...
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### Checking positive semidefiniteness in MATLAB

Let $\mathbf{A}$ be a $n\times n$ matrix. I want to check in MATLAB if it is PSD or not. Which tests, in MATLAB, should I do for this purpose? I know that if $\mathbf{A}$ is PSD then following holds ...
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### Is there a decomposition $U U^T$?

We know that there exist the Choleski decomposition $M = L L^T$ where $M$ is a positive definite matrix and $L$ a lower triangular one. Does it exist a similar decomposition in $M = U U^T$ ...
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### To what extent are the Jordan-Chevalley and Levi Decompositions compatible.

I know that the Jordan-Chevalley decomposition for real Lie algebras only applies to semisimple Lie algebras, but in general the addititive J-C decomposition says that for ANY operator, we can ...
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### Find rank of the special block symmetric and persymmetric matrix

I meet a difficult problem recently. The problem is to find the rank of a special matrix: $$X = \begin{bmatrix} S & P \\ P & JSJ\end{bmatrix}\in \mathbb{R}^{2m\times2m} ,$$ where $S$ is ...
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### How to find the inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ without forming the Kronecker product?

Is there a good way to compute the inverse of Inverse of $A\otimes A + (I+D\otimes D)^{-1}(D\otimes D)$ that doesn't require forming the full Kronecker product? Here $A$ is symmetric, positive ...
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### Papers on Tensor factorization

I started reading a few papers on Tensors. Right now I am reading a paper by Chi and Kolda called “On Tensors, Sparsity, and Nonnegative Factorizations”. I passed linear algebra and calculus courses. ...
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### A set of four matrix equations

Let $\{\mathbf b_1,\mathbf b_2,\mathbf b_3\}$ be a basis for $\mathbb R^3$ and consider three arbitrary vectors $\mathbf w_i=\sum_j\omega_{ij}\mathbf b_j$. We define the following two $3\times 9$ ...
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### Volume of subgroups and coset decompositions

Let $F$ be a number field, $\mathfrak{o}$ be its ring of integers and $\mathfrak{p}$ be the associated maximal ideal. Consider the subgroup $H$ of ${GL}(3, \mathfrak{o})$ given by matrices of the ...
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### Singular Value Decomposition for Rectangular Matrices

In the book Linear Algebra Done Right (Axler, 237), the following is the statement of the theorem that gives us the existence of a singular value decomposition: 7.51 $\qquad$ Singular Value ...
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### How to deblur a image matrix blured by two circulant matrix?

We suppose an image matrix $X\in \mathbb{R}^{n_1\times n_2}$ is blurred by two circulant matrices $\Phi_1 \in \mathbb{R}^{n_1\times n_1},\Phi_2 \in \mathbb{R}^{n_2\times n_2}$. We can observe the ...
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### Simplify $x^TA(AA^T+I)^{-1}A^Tx$

I need to simplify the term $$x^TA(AA^T+I)^{-1}A^Tx$$ where $A$ is $N\times n$ matrix and $x$ is a $N\times 1$ vector. I have some information of $x^Tx$. Could anyone know how to simplify the term ...
### Show that partial pivoting leads to an $LU$ decomposition of $PA$.
Exercise: show that for every non-singular matrix $A$, partial pivoting leads to an $LU$ decomposition of $PA$ so: $PA = LU$. I have the following theorems I can use: Theorem 1: Assume that the ...