Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
2,544
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How is the reduced matrix of a matrix?
I'm trying to generate the characteristic polynomial of the matrix below, from the reduced matrix. But the calculations are not correct, because, probably, my reduced matrix is wrong. I hope this ...
- 19
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11
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Constrained Hankel matrix decomposition
I want to decompose a square Hankel matrix $\bf {H}$, whose elements below the anti-diagonal are zeros. The decomposed factors should necessarily meet the following constraints:
$$\begin{equation} \bf ...
- 27
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SVD of rank-$1$ tall matrix
I am learning the singular value decomposition (SVD) and there has been some moments when I've been unable to fill out the vectors in $\bf U$ and I'm not sure what I'm meant to be doing for ${\bf A} = ...
- 31
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Complex Schur Decomposition proof Van loan
Lemma 1:
Let $A \in \mathbb{C}^{n \times n}$, $B \in \mathbb{C}^{p \times p}$, and $X \in C^{n \times p}$ satisfy
$
AX=XB , \ \ \ rank(X) = p,
$
then there exists a unitary $U \in \mathbb{C}^{n \...
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Case of equality in spectral norm matrix triangle inequality
Let $(A,B)\in M_n(\mathbb{R})\times M_n(\mathbb{R})$ be two matrices. We denote by $\|\cdot\|_2$ the spectral norm.
Without any additional assumptions on $A$ and $B$, can we characterize the case of ...
- 49
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23
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Complex Schur decomposition
The complex Schur decomposition goes as follows:
For all $A \in \mathbb{C}^{n \times n}$ there exists a unitary matrix $U$ such that $U^{*}AU$ is triangular say, $U^{*}AU=T$. I have seen it being ...
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Showing that $GL(2,p) = SL(2,p) \rtimes H$
I want to show that $GL(2, p)$ is the semidirect product of $SL(2, p)$ and the subgroup $H$ of $2 \times 2$ diagonal matrices with elements on the diagonal being $1$ and $\alpha$, where $\alpha \in \...
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Is there an example that a specific matrix has Jordan block $J_{2}(i)$?
Consider a matrix of the form $\begin{bmatrix}
A & C\\
-C^{T} & B
\end{bmatrix}$ where A and B are symmetric matrices.
Can matrices of this type have a Jordan normal form representation ...
- 11
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19
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Compare the condition number for the least square problem matrix
I am working on a problem with the following matrix $$G=\begin{pmatrix}I&A\\
A^T&0 \end{pmatrix}$$ where $A\in\mathbb{R}^{m\times n}$ and $m>n$ with full column rank.
Then by rescaling, ...
- 381
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If B is a $2\times3$ matrix with some condition, find a non-zero matrix $C$ such that $BCB^T=0$
The following is the Question:
Consider a $3 \times 3$ real matrix $A=\left(\begin{array}{ccc}1 & -2 & 0 \\ -2 & 5 & -1 \\ 0 & -1 & 2\end{array}\right)$.
(a) Show that $A$ is ...
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9
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Matrix product decomposition
Consider a square real non-negative matrix $\mathbf{A}$.
My question is: is there a way to write $\mathbf{A}^{t}\big({\mathbf{A}^{t}}^{T}\big)$ (where $^T$ is transpose and $^t$ is matrix t_th power) ...
- 127
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1
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Proof of existence Schur decomposition
I was reading proof of existence of Schur decomposition.
I understand everything except one thing.
Why submatrices like $A_2$ have same eigen value with main matrix $A$ like $\lambda_2$ ?
A ...
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Find 3x3 Matrix with specific eigenvalue and eigenvector
I have to find a 3x3 matrix M with only one given eigenvalue λ = 0 and an eigenvector ...
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Can I decompose this matrix into separate parts?
Right now I have a matrix of the form
\begin{pmatrix}
0 & (a^\top b) ~ b ~c^\top \\
c ~ b^\top (b^\top a) & 0,
\end{pmatrix}
where $a$ and $b$ are vectors of the same dimension, and $c$ is ...
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QZ decomposition of singular matrices - Multiplicity of $\lambda = 0$
Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be $n \times n$ singular matrices with the same eigenvalues (including multiplicity). Consider that the multiplicity of the eigenvalue $\lambda = 0$ of $\...
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Order of eigenvectors within basis for Jordan Normal Form?
I'm currently baffled as I thought that the order of eigenvectors within the basis of a JNF decomposition doesn't matter. I may have a made a mistake in my working, but if not, is there a general rule ...
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Writing a given matrix as a product of fewest elementary matrices
Write the matrix $$\begin{bmatrix} 1 & 2 \\\ 3 & 4 \end{bmatrix}$$ as a product of elementary matrices, using as few as you can, and prove that your expression is as short as possible.
I ...
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Polar Decomposition using Pseudoinverse
It is well known that every operator $A$ on a finite-dimensional space has a polar decomposition $A = UP$ where $U$ is a unitary operator and $P = \sqrt{A^*A}$. Further, when $A$ is not invertible the ...
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Properties of $3\times 3$ matrices
Given two sets of corresponding points $\mathbf{P}_1, \mathbf{P}_2 \in \mathbb{R}^{d \times n}$, where $\mathbf{P}_1, \mathbf{P}_2$ are pointclouds expressed as matrices, we can derive a matrix $\...
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Constructing an arbitrary set of real vectors whose pairwise dot products are bounded
I am trying to write a code that generates an arbitrary set of real $n$-dimensional vectors, whose all pairwise dot products are greater than some lower bound $l$ and less than some upper bound $u$. ...
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Integer representation of rectangular matrices
Given a $n\times m$ rectangular matrix A over a field $k$, does there exist a $m\times q $ matrix $Q$ such that $QQ^T=I$ and $AQ$ has only integer elements.
This does not hold in general. A simple ...
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Eigendecomposition with rectangular matrices
I have a matrix $M = ABC$ where:
$A$ is an $n \times m$ matrix. ($n$ rows and $m$ columns)
$B$ is an $m \times m$ diagonal matrix, and all its diagonal elements are positive.
$C$ is an $m \times n$ ...
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25
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About a definition of "rectangular generalized invariant matrix" in a paper
Definition 2.16 in the paper "Universality of Approximate Message Passing Algorithms and Tensor Networks" (PDF link via arXiv.org) by Wang, et al., describes a rectangular generalized ...
- 490
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15
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Separable convolution of two summed 2D Gaussian kernels
I am performing a convolution of a dataset with a large 2D kernel which is defined as the sum of two Gaussians in the following form
$$
f(r) = \exp(- r^2 / \sigma_1^2) + \beta \cdot \exp(- r^2 / \...
- 11
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1
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50
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LU factorization of a non-square matrix
I am currently working on LU factorization of a matrix, and I'm struggling with the following problem:
$$A=\begin{bmatrix}1&2&3&1&2\\ 1&4&2&3&1\\ 2&2&-1&1&...
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2
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SVDs of the matrix $\begin{bmatrix} 1 & 1 \\ 0 & 0\end{bmatrix}$
This is from Trefethen's Numerical Linear Algebra, lecture 4, exercise 1(d).
Determine the SVDs of the matrix
$$\begin{bmatrix} 1 & 1 \\ 0 & 0\end{bmatrix}$$
This matrix maps both $[1, 0]$, $...
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Is there a name for this kind of matrix decomposition?
A square matrix $\mathbf{A}$ (of size $a\times a$) is not full rank and I want to decompose it as $\mathbf{A}=\mathbf{B}\mathbf{B}^\top$ where $\mathbf{B}$ is an $a\times(a-1)$ matrix of the $a-1$ ...
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Can a LDL decomposition be efficiently converted into UDU decomposition?
Question:
If you are given matrices $L,D_L$ where
$L$ is a lower unit triangular matrix, $D_L$ is a diagonal matrix, and $L,D_L$ are a LDL decomposition of a particular symmetric positive definite ...
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1
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Repeated Matrix Approximations / Approximation of Smaller Rank
I am attempting to prove a fact about matrix approximations using the singular value decomposition.
For a given matrix $M$, we denote the rank $r$ approximation of $M$ by $A(r, M)$. I am trying to ...
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How to decompose a matrix into a sum of circulant matrices
I have a very particular matrix
$A = \alpha e_1 e_1^T$, where $\alpha$ is constant, means it has only one element in the diagonal that is different from zero.
Can this matrix be decomposed into a ...
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Inequality on Frobenius norms of the difference of 2 matrices
Can I write $$|| V D_1 D_2 V^T -Q D_1 D_2 Q^T||_F \le f(D_1,D_2)_F || V D_2 V^T -Q D_2 Q^T||_F $$
for some $f$? I have that $V$ and $Q$ are orthonormal matrices, $D_1$ and $D_2$ are diagonal matrices....
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How to obtain this simplification
I was studying a linear algebra problem and I encountered this step in the solution :
\begin{aligned}
& \mathbf{1}^T\left(\mathbf{1 1}^T+I \sigma_v^2\right)^{-1} y =\left(\mathbf{1}^T \mathbf{1}+\...
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Characterize the max and min eigenvalues of a positive definite matrix under congruence transform with semi-definite matrix
I am looking to characterize the maximum or minimum eigenvalues of a matrix $B = A Q A + \mathbf{I}-A$ where
$A$ is positive semi-definite, symmetric, and has eigenvalues $\lambda_i\in[0,1)$
$Q$ is ...
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0
answers
51
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Eigenvalue bounds on $2\times 2$ block matrix with diagonal off-diagonals
Let $A,D\in\mathbb{C}^{n\times n}$ with eigenvalues $\alpha_i,\delta_i$ and let $b,c\in\mathbb{C}$. Without loss of generality, I've been able to prove the following statements.
The eigenvalues of $K ...
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Eigenvalues of quasi circulant matrix
I have to find the eigenvalues of a quasi-circulant matrix. I found this answer here: Eigenvalues and eigenvectors for a quasi-circulant matrix
But I cannot understand what is $nt$ and how the guy ...
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Any quantitative methods of eigenvalues and matrix compression?
I have recently been studying how to use eigenvalues and eigenvectors to compress matrices. Specifically, we have a positive semi-definite matrix $A$, which can be diagonized by an orthogonal ...
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Eigenvalue decompositin equal to singular value decomposition
Question: when is the eigenvalue decomposition of a matrix equal to its singular value decomposition?
Answer: when A is hermitic and has positive eigenvalues.
I don't really understand. If we have A=A*...
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Fast algorithm for incremental randomized SVD
I have a list of covariance matrices $\{\Sigma_i\}$. I want to be able to take the (randomized for performance) SVD of the average of different (incremental) subsets of this list in order to perform a ...
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Matrix Decomposition as $M=AA^\intercal$
I am interested in decomposing a square $n\times n$ matrix $M$ with positive integer elements as a square matrix $A$ with positive integer elements and such that $M=AA^\intercal$. I have a couple of ...
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Diagonalization of symmetric matrices of functions
Let $U\subseteq\mathbb{R}^m$ be an open subset such that $0\in U$ and $n\leq m$. Let
$G(p)=\begin{pmatrix}
g_{11}(p) & g_{12}(p) & \cdots & g_{1n}(p)\\
g_{12}(p) & g_{22}(p) & \...
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Find eigenfunctions with kronecker product and sum
I need to find the eigenvalues of
$C+D$, where C is an nxn circulant matrix, and D is an nxn diagonal matrix. I know the eigenvalues $\lambda_i$ of $C$ and the eigenvalues $\mu_i$ of D. However, these ...
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How to decomposition a Hermitian matrix subtracted from Identity?
I've got a complex vector$A\in \mathcal{C}^{M\times 1}$, where$A^H$means the conjugate transpose of $A$, and $A^{-1}$ means the inverse or pseudo-inverse of $A$. The target matrix is $X$.
$X=I-A(A^HA)^...
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If $M\in M_{m \times n}(\mathbb{R})$ is a rank $r \geq 1$ matrix then it can be written as a sum of exactly $r$ rank 1 matrices
I have problem with the proof of the following statement.
If matrix $M\in M_{m \times n}(\mathbb{R})$ has rank $r \geq 1$, then it can be written as a sum of exactly $r$ rank-$1$ matrices
I try in ...
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The matrix inverse of $\begin{pmatrix} a_{11} 10^{10}& a_{12} 10^{30} \newline a_{21} 10^{20}& a_{22} 10^{40} \end{pmatrix} $ and an algebra?
The question started with how to compute the matrix inverse with a scale separated orders.
A related question could be found here:
Compute matrix inverse at scale separated orders
This is, in $2\times ...
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1
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If the symmetric part of a matrix is PSD then the matrix is PSD [duplicate]
My professor said this during my linear algebra class. But I am not sure how to prove this.
If A is any matrix, then it can be decomposed into a symmetric and anti-symmetric part like below-
$$
A = \...
- 87
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votes
1
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31
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Square root of PSD matrix and unitary freedom
Consider a given PSD matrix $S$, which can be written as:
$$S= A A^\top$$
where $A$ is its unique PSD square root. I am computing $A$ through the EVD of $S= U DU^\top$, i.e., $A= U D^{1/2}U^\top$.
I ...
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22
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QR decompositon and Q is upper triangular
Suppose $A\in\mathbb{R}^{n\times n}$ and $D=\text{diag}\{d_1,\cdots, d_n\}\in\mathbb{R}^{n\times n}$. Construct an orthogonal $Q$ such that $$Q^T A-DQ^T=R$$ is upper triangular.
I want to construct $Q$...
- 381
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Best inverse / minimization solution of ill conditioned matrix and underdetermined system
I have an ill-conditioned matrix G representing Green's function estimates of a strain model for a rock mass that has been instrumented with optical fiber measuring strain. My problem is seemingly the ...
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0
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14
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Is it valid to rely on a derived matrix for interpolation/extrapolation?
For example, suppose you have a known $\mathbf{A}$ and a known $\mathbf{Y}$ with an unknown $\mathbb{X}$:
$$
\mathbf{A}\mathbb{X} = \mathbf{Y}\\
\mathbb{X} = \mathbf{A^{-1}Y}
$$
Now, suppose you ...
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0
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How are a matrix, a result matrix of a couple of matrix multiplies, the inverse of that matrix, and doing the matrix multiplies backwards connected?
The matrices I use are for the canvas in webbrowsers. https://developer.mozilla.org/en-US/docs/Web/API/CanvasRenderingContext2D/transform
where [a,b,c,d,e,f] is ...
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