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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Computing the Smith Normal Form
Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix
$$R := \begin{bmatrix}
-6 & 111 & -36 & 6\\
5 & -672 & 210 & 74\\
0 & -255 &...
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6
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Proof that $\text{det}(AB) = \text{det}(A)\text{det}(B)$ without explicit expression for $\text{det}$
Overview
I am seeking an approach to linear algebra along the lines of Down with the determinant! by Sheldon Axler. I am following his textbook Linear Algebra Done Right. In these references the ...
- 1,461
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votes
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answer
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When does a Square Matrix have an LU Decomposition?
When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do ...
- 411
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LU Decomposition vs. Cholesky Decomposition
What is the difference between LU Decomposition and Cholesky Decomposition about using these methods to solving linear equation systems?
Could you explain the difference with a simple example?
Also ...
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votes
7
answers
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Understanding the singular value decomposition (SVD)
Please, would someone be so kind and explain what exactly happens when Singular Value Decomposition is applied on a matrix? What are singular values, left singular, and right singular vectors? I know ...
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Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal. Must $A$ be diagonal?
Let $A=(a_{ij})$ be an $n\times n$ matrix. Suppose that $A^2$ is diagonal? Must $A$ be diagonal.
In other words, is it true that
$$A^{2}\;\text{is diagonal}\;\Longrightarrow a_{ij}=0,\;i\neq j\;\;?...
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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 ...
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}...
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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 ...
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