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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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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 operations ($P^{-1}DP$) on a vector: Rotation…
user541686
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How unique are $U$ and $V$ in the singular value decomposition $A=U\Sigma V^\dagger$?

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 $V$ are not). My question is, are $U$ and $V$…
capybaralet
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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 school course on Linear Algebra last year, but…
Sidd Singal
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Proof that $\text{det}(AB) = \text{det}(A)\text{det}(B)$ without explicit expression for $\text{det}$

Overview I am seeking an approach to linear algebra along the lines of Down with the determinant! by Sheldon Axler. I am following his textbook Linear Algebra Done Right. In these references the author takes an approach to linear algebra that avoids…
Jagerber48
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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 precisely, the last two columns of Q) gives the null space: $$Q=…
whitegreen
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Computing the Smith Normal Form

Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $$R := \begin{bmatrix} -6 & 111 & -36 & 6\\ 5 & -672 & 210 & 74\\ 0 & -255 & 81 & 24\\ -7 & 255 &-81 & -10 \end{bmatrix}$$ Reduce this matrix using…
Euden
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Understanding the singular value decomposition (SVD)

Please, would someone be so kind and explain what exactly happens when Singular Value Decomposition is applied on a matrix? What are singular values, left singular, and right singular vectors? I know they are matrices of specific form, I know how to…
Celdor
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When does a Square Matrix have an LU Decomposition?

When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do we tell if it does/doesn't exist? (Note:…
Highrule
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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 could you explain the differences between these…
mertyildiran
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Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal. Must $A$ be diagonal?

Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal? Must $A$ be diagonal. In other words, is it true that $$A^{2}\;\text{is diagonal}\;\Longrightarrow a_{ij}=0,\;i\neq j\;\;?$$
Medo
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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 not sure — reflection, shear, scaling and…
Sunny88
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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 decomposition? Thank you.
Gatsu
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Why is there not a test for diagonalizability of a matrix

Let $A$ be square. This question is a bit opinion-based unless there is a technical answer. I think it is helpful tho. Also, this question is closely related to this question : quick way to check if a matrix is diagonalizable. Based on the answers,…
Rias Gremory
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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 when one already knows how to compute $A^{-1}$ and…
user856
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How to understand the spectral decomposition geometrically?

Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = \sum_{i=1}^k\frac{1}{\lambda_i}e_ie_i'$$ How to understand spectral decomposition and…
Jill Clover
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