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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.

In linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

  • For instance, when solving a system of linear equations $Ax=b$, the matrix $A$ can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix L and an upper triangular matrix U.

  • Similarly, the QR decomposition expresses $A$ as QR with Q an orthogonal matrix and R an upper triangular matrix.

Other decomposition techniques include: Block LU decomposition, LU reduction, rank factorisation, Cholesky decomposition, etc.

Source: Wikipedia.

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Is this the correct iterative version of the multiplicative update rule for matrix factorization?

So we have $A_{m\times n}\approx P_{m\times k}Q_{k\times n}.$ Using the multiplicative update rule due to Lee we, in general, write that: $$P\leftarrow P \circ \frac{AQ^T}{PQQ^T}$$ and $$Q\leftarrow Q \circ \frac{P^TA}{P^TPQ}.$$ Since I am…
Student
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Complex factorisation of a psd matrix

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. Assume that $A=L\cdot L^{t}$ for $L\in\mathbb{C}^{n\times m}$. Can we infer that $L$ is intact real, i.e., has only real entries?
user382144
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Proof that in the Jordan-Chevalley decomposition of an invertible matrix, the diagonalisable matrix is invertible

Let $F$ be a field and $n\in \mathbb N^*$. The (additive) Jordan-Chevalley decomposition theorem states that for any matrix $M\in \mathcal M_n(F)$ whose characteristic polynomial splits into linear factors there exists a unique decomposition $M=D+N$…
James Well
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I need to find $v$ so that $v\times v'$ equals $A(n \times n)$

Be $A_{n\times n}$ symmetric. I need to find $v_{m\times n}$ so that $v^T \times v = A$ for an algorithm in Julia. Can anyone help me (either w/ Julia or linear algebra)?
Gabriel Milan
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Eigenvalue decomposition of Cross Gramian matrix

For a symmetric state space system $G(s)=\left\{A,B,C,D\right\}$, the cross Gramain matrix $R$ is the solution of $$AR+RA+BC=0$$ Using eigenvalue decomposition, problem is to obtain a matrix U which diagonalizes the cross gramian matrix $R$,…
Dee Kay
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Non-Negative Matrix Factorization Constraints

$A\in{\mathbb{R}^+}^{m\times n}$ (i.e. a matrix with elements that are positive real numbers). The rank of $A$ is $k$, and $k < min(m,n)$. Therefore $A = BC$, where $B\in{\mathbb{R}^+}^{m\times k}$ and $C\in{\mathbb{R}^+}^{k\times n}$. Also, we…
Alex
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What is a practical real world problem that requires the usage of matrix factorization

I am not asking about the various methods of decomposing a matrix. I am looking for a real world problem that can be represented in matrix form and then can be solved using the concepts of decomposition. i.e in short, why do we need to learn matrix…
user105947
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Which row/column in a matrix genarate which singular value

I know that svd of a matrix arranges the singular values in descending order. My question is: Is it possible to know which row/column generate a specific singular value, for e.g. the largest one. Thanks in advance.
Tom
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A question about positive definite matrix decomposition

G and V are two positive definite symmetric matrices. How to find a symmetric matrix W such that: $$W G W =V$$
xjtein
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