# 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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### Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?

I'm trying to intuitively understand the difference between SVD and eigendecomposition. From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three basic ...
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### How unique are $U$ and $V$ in the singular value decomposition $A=UDV^\dagger$?

According to Wikipedia: A common convention is to list the singular values in descending order. In this case, the diagonal matrix $\Sigma$ is uniquely determined by $M$ (though the matrices $U$ and ...
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### How can you explain the Singular Value Decomposition to non-specialists?

In two days, I am giving a presentation about a search engine I have been making the past summer. My research involved the use of singular value decompositions, i.e., $A = U \Sigma V^T$. I took a high ...
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### How is the null space related to singular value decomposition?

It is said that a matrix's null space can be derived from QR or SVD. I tried an example: $$A= \begin{bmatrix} 1&3\\ 1&2\\ 1&-1\\ 2&1\\ \end{bmatrix}$$ I'm convinced that QR (more ...
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### Is it true that any matrix can be decomposed into product of rotation, reflection, shear, scaling and projection matrices?

It seems to me that any linear transformation in ${\Bbb R}^{n \times m}$ is just a series of applications of rotation — actually i think any rotation can be achieved by applying two reflections, but ...
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### Relation between Cholesky and SVD

When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed). Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
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### The benefit of LU decomposition over explicitly computing the inverse

I'm going to teach a linear algebra course in the fall, and I want to motivate the topic of matrix factorizations such as the LU decomposition. A natural question one can ask is, why care about this ...
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### Cholesky decomposition of the inverse of a matrix

I have the Cholesky decomposition of a matrix $M$. However, I need the Cholesky decomposition of the inverse of the matrix, $M^{-1}$. Is there a fast way to do this, without first computing $M^{-1}$? ...
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### 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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### A criteria for Jordan decomposition of a matrix over a general ring

When I look at different proofs of the Jordan-Chevalley decomposition of a matrix, the minimal hypothesis I usually found is about the perfection of the field over which such decomposition occurs (e.g....
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### How to prove the existence and uniqueness of Cholesky decomposition?

Given a real Hermitian positive-definite matrix $A$ is a decomposition of the form $A=L L^T$ where L is a lower triangular matrix with positive diagonal entries. I read some proofs about the existence ...
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### If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$

Prove that if $A$ is a square matrix and $\mathrm{Tr}(A)=0$, then there exists an invertible matrix $R$ such that the matrix $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$. It seems ...
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### Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidefinite matrix $\Theta$. However, I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the ...
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### A normal matrix with real eigenvalues is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
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### Solving a linear system with Cholesky factorization

So, I've reached the following problem: Given $A \in R^{nxn}$ positive-definite and symmetrical, $B\in R^{nxn}$ and the vector $c\in R^n$ write an algorithm to solve to following situation: $Ax=Bc$ ...
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### Polar decomposition of real matrices

Can we decompose real matrices in a way that is analogous to polar decomposition in the complex case? What I mean is: given an invertible real matrix $M$, can we always write: $$M = OP,$$ maybe ...
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### What is the Cholesky Decomposition used for?

It seems like a very niche case, where a matrix must be Hermitian positive semi-definite. In the case of reals, it simply must be symmetric. How often does one have a positive semi-definite matrix in ...
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