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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Decompose a $3 \times 3$ orthogonal matrix into a product of rotation and reflection matrices
It holds that every orthogonal matrix can be expressed as a product of rotation and reflection matrices.
We can prove that every $2 \times 2$ orthogonal matrix can be represented in the form
$$\begin{...
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What happens when we don't impose any structure on constituent matrices when performing matrix decomposition?
$$
\newcommand{\mat}{\mathbf}
$$
In SVD, LU, QR, and other decompositions we impose constraints on the constituent matrices for various interpretability and/or computational convenience reasons. For ...
15
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Why is there not a test for diagonalizability of a matrix
Let $A$ be square.This question is a bit opinion based, unless there is a technical answer.I think it is helpful tho. Also this question is closely related to this question : quick way to check if a ...
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DFT in Galois Field [closed]
I have a $n \times n$ circulant matrix A over a Galois Field $GF(2^5)$. Can I use the similar logic as Discrete Fourier Transform (DFT) to write it as $A=FDF^{-1}$ ? I am asking this because I need to ...
1
vote
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Any decomposition of inverse of nonnegative diagonal matrix times a PSD matrix plus lambda times Identity?
I generally have to solve the following system:
$$
(DA + \lambda I)^{-1} v
$$
where $D$ is a diagonal matrix with nonnegative entries, $A$ a symmetric, positive semi-definite (PSD) matrix, $I$ is the ...
- 133
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Classifying maps of finitely generated abelian groups up to automorphism
We have a nice characterization of f.g. abelian groups that factors them into cyclic groups. A homomorphism between them is just a matrix of maps $\mathbb{Z}_{p^i} \to \mathbb{Z}_{p^j}$ between the ...
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Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?
I know in general, matrix multiplication is not commutative, but would it be true in this special case?
$D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
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Singular value of a bidiagonal matrix?
Consider a $n\times n$ matrix:
\begin{equation}
X=\begin{bmatrix}
a &1-a & & \\
& a &1-a & \\
& & \ddots &\\
& & & 1-a\\
& & & a
\end{...
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What is the square root of a square matrix squared? [closed]
Admittedly, made the title a little funny, but this is a valid question.
I have come across the following equation
$$
I x^2=AA
$$
where $I$ is a unit matrix, $A$ is a square matrix of the same ...
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Suitability of QR factorization for solving a ill-conditioned linear system.
I am trying to solve a linear system $Ax = b$ where
$$
A =\begin{bmatrix}
2& 9& 2& 1& 4& 1& 0& 0& 0 \\
9& 65& 9& 1& 4& 1& 0& 0&...
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Compressed image using SVD draws a clear line between part that's blank and part with a drawing. Why? [closed]
I'm trying to compress grayscale images using SVD. This is the original image:
Yes, there's a lot of blank space.
I then choose the x% largest singular values, perform the transformed matrices ...
0
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1
answer
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For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?
Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows:
$$...
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What is the probability that a random square matrix is eigendecomposable? [duplicate]
Suppose I construct an $n \times n$ matrix with all entries given randomly in (-1, 1). What is the probability that such a matrix has a full set of $n$ eigenvectors; that is, it is eigendecomposable?
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Singular values on streching the vectors
For the following statement: for a vector $x$ and a matrix $A$, if the vector $x$ is not in the null space of $A$, the vector $x$ will at least be stretched by the smallest non-zero singular value, i....
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For a symmetric matrix $B$ and following four relevant matrices $P,Q,C,D$, what's the relation between $QP$ and $CDC^{\rm T}$?
Suppose $B\in\mathbb{R}^{n\times n}$ is a symmetric matrix with ${\rm rank}(B)=r$. Then $B$ is equivalent to $\tilde{B}$ in (1), where $I_{r}$ denotes the identity matrix of order $r$. That is, there ...
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