Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
2,713
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How to estimate the inverse of a non-invertible matrix?
So I'm working on a machine learning problem where my solution requires taking the inverse of a matrix at some point. The problem is that this matrix is sometimes non-invertible. In theory the the ...
- 11
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2
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For any SVD $A = U\Sigma V^T$ of a positive definite, symmetric matrix $A \in \mathbb{R}^{n \times n}$, we have $U = V$.
First of all, I've read through all of the answers here and here, but neither of those threads was able to give completely satisfying answers.
Now, I understand that, if $A$ is symmetric and positive ...
- 23
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34
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need a Jacobian of an orthonormal basis for the column space of a matrix $\mathbf{X}$
I require to find a linear decomposition of any tall matrix $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$, where $M \geq N$, into some uniquely defined decomposition $\mathbf{X}$ $=$ $\mathbf{A}$ $\...
- 41
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Relationship between BCH code and asymmetric Ramanujan bipartite graph ( possibility for a research collaboration)
I have been working on a research topic that deals with the binary matrices arising from the BCH codes by selecting code vectors of specific weight while discarding the rest of the code vectors that ...
- 49
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1
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32
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Exponeintal of symmetric triangular matrix
I want to know the exponeintal of given $n \times n$ symmetirc real tridiagonal matrix ${\bf K}_n$, which is defined as
$${\bf K}_n=\begin{bmatrix}
0 & a & 0 & 0 & \dots & 0 & ...
- 530
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votes
1
answer
52
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Showing existence of symplectic transformations preserving a quadratic form
Question: I need help to prove the following statement.
Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...
- 108
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Optimization of eigenvalue of matrix with discrete variables
Suppose we have a matrix $H$ like this, where $H_{ii}$ is a discrete variable (for example out of $[1,2,3]$) and $H_{ij}$ is a value that depends on the two adjacent values ($H_{ii}$ and $H_{jj}$). $H$...
- 31
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28
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Woodbury matrix identity with a minus sign
Is there a form of Woodbury matrix identity
$(A + UCV )^{-1} = A^{-1} - A^{-1}U (C^{-1} + VA^{-1}U )^{-1} VA^{-1}$
But with a minus sign? i.e. $(A - UCV )^{-1}$ It seems like I have to painfully ...
- 1,634
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Proving that the rank of the following matrix is $6$.
In my research work I have come across a matrix which has the rank equals to $6$. I begin defining my problem as follows: Let $P \in \{0,1\}^{7 \times 7}$ denote the right shift matrix defined by
$ P =...
- 49
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Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_{min}$, of $A$?
What I want is something like: $\sigma_{min}$...
- 115
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3
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81
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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{...
- 137
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3
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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 ...
- 477
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1
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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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votes
1
answer
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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 ...
- 4,867
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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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1
answer
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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{...
- 304
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votes
1
answer
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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 ...
- 101
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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
$$
A =\begin{bmatrix}
2& 9& 2& 1& 4& 1& 0& 0& 0 \\
9& 65& 9& 1& 4& 1& 0& 0&...
- 1,261
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Compressed image using SVD draws a clear line between part that's blank and part with a drawing. Why? [closed]
I'm trying to compress grayscale images using SVD. This is the original image:
Yes, there's a lot of blank space.
I then choose the x% largest singular values, perform the transformed matrices ...
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For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?
Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows:
$$...
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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 ...
- 155
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How to decompose a simple $3\times3$ shear transformation into a rotation, scale, and rotation
Is there a simple way to decompose the following $3\times3$ shear matrix into the product of a
rotation, (non-uniform) scale, and another rotation? Or Perhaps
some other combination of rotations and ...
- 802
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0
answers
72
views
Find one quartic root of a matrix
I have found the previous spectral decomposition of the matrix $$A=\begin{pmatrix} 1 & 1 & 0 \\
0&1&1\\
1&0&1
\end{pmatrix}.$$
You can see I verified such decomposition indeed ...
- 1,494
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A low-rank approximation problem with rank constraints
I am seeking a solution or some ideas to address the following problem:
$$
\begin{aligned}
&\text { minimize }_{\widehat{A}, \widehat{B}} \quad\|A-\widehat{A}\|_2 + \|B-\widehat{B}\|_2 \\ &\...
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votes
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Completely non- normal matrix
Let $M_n(\mathbb{C})$ be the space of $n \times n$ matrices with complex entries. A matrix $N$ is said to be normal if $N^*N=NN^*$ where $N^*$ denotes the conjugate transpose of $N$. One can think of ...
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Orthogonal projection of a complex valued matrix onto the space of Hermitian matrices
It is well known that any real matrix $A$ can be decomposed as the sum of a symmetric and a skew-symmetric matrix as follows:
$$
A= \frac{A+A^T}2+\frac{A-A^T}2.
$$
The decomposition is orthogonal, ...
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Detect linearly dependent columns from a full-row rank matrix.
Let $A\in \mathbb{R}^{m\times n}$, with $m\leq n$, be a full-row-rank matrix. Then, there exist a collection of $n-m$ columns in $A$ that are linearly dependent on the other columns. What is the most ...
- 372
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2x2 blocks in the QZ algorithm
How are the $2\times2$ blocks supposed to be diagonalized in the QZ-Algorithm? Taking the matrix pencil (A,B) and finding it's generalized Hessenberg decomposition (H,R) for which $\exists Q,Z \in \...
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1
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Matlab qz algorithm not reliable
I programmed my own version of the qz algorithm and, while testing it's results using the matlab qz algorithm, I found a particular case where my solution reaches an upper-triangular matrix and matlab ...
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Is there a closed form solution?
Question:
Let $A\in\mathrm{GL}(n,\mathbb{R})[[u]]$ with $A=A^T$, that is a symmetric real matrix, which can not be decomposed into a blockmatrix with $0$ blocks, such that
$$ A = \begin{pmatrix} A_1 &...
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Degrees of freedom of an $r$-ranked tensor?
I'm trying to determine the degrees of freedom for parameters of a tensor in shape $(J_1,\dots,J_D)$ and with rank $r$, where "rank" refers to the smallest number of rank-1 tensors whose sum ...
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Inverse and Determinant of Matrix $Axx^TA+cA$
Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
- 2,497
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If I can solve $Ax = b$ efficiently, is there a method to solve $(I+A)x=b$ efficiently?
Say if I already have the LU factorization of a square matrix $A$, is there an efficient way to get the LU factorization of $I+A$? (We may assume all the matrix I mentioned is invertible.)
I know from ...
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Best method for sequential small size Hermitian smallest eigenpair problem
I am working on a perhaps rather strange problem. To find the smallest eigenvalue and its eigenvector, for a large number (a few billions) of small (20 * 20 to 200 * 200) Hermitian matrices. These ...
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If $A^*A=A^*B=B^*A=B^*B$, prove that $A=B$.
Suppose that $H$ is a Hilbert space and $A,B\in B(H)$ are such that $A^*A=A^*B=B^*A=B^*B$,then $A=B$.I'm having difficulty trying to read a proof that solves this problem.
First, the author decomposes ...
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Fast way to compute the largest eigenvector of an expensive-to-compute matrix
Consider an $N \times N$ Hermitian positive semidefinite matrix $M$. Computing the elements of $M$ is expensive so we wish to compute as few as possible. We can assume that $M$ can be approximated as ...
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4
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$O$ orthogonal with $\det(O)=-1$ implies $||\Omega - O \Omega O^{T}|| = 2 $?
Let $O\in \mathrm{O}(2n)$ be an orthogonal matrix. Let $\Omega$ be the matrix $\Omega:=
\bigoplus^n_{i=1} \begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix}.$
Is it true that:
$\det(O)=-1$ ...
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Quadratic form of a matrix where non-standard decomposition is known
Let $1\le m<<n$ for integers $m,n$. I have two symmetric matrices of size $n\times n$, say $A$ and $B$. My goal here is to know a simple form, or an upper bound on the following quadratic form:
$...
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Minimize $||A-AWW^TA^T||_F$ w.r.t. $W$
Given $n \in \mathbb{N}$ and $A \in \{0,1\}^{n \times n}$, we aim to find
$$\arg \min_{W \in \mathbb{R}^{n \times n}} f(W) = ||A-AWW^TA^T||_F,$$
where $||\cdot||_F$ represents the Frobenius norm with
$...
- 2,150
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Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.
The problem is a conituation of this problem, but over finite fields.
In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
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Euclidean norm of a vector resulting from a matrix multiplied by an orthonormal matrix multiplied by an arbitrary vector!
Let $A \in \{0,1\}^{m\times n}$ be a binary matrix with $ m < n$, and let $U \in \mathbb{C}^{n \times n}$ be an arbitrary orthonormal matrix. Let $\sigma_{min}$ denote the smallest non-zero ...
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votes
1
answer
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views
Binary matrix power for a specific entry.
Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$.
I am interested in the following decision procedure:
Given a ...
- 141
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27
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Understanding QR decomposition in the context of a cubature Kalman filter
I am working on implementing a square-root cubature kalman filter based on the algorithm presented in this conference paper, as well as this paper more broadly.
I have got the algorithm to run; ...
0
votes
0
answers
18
views
Finding Correspondence between Matrices after Decomposing a Hermitian Matrix
After decomposing the Hermitian matrix $M$, I obtain a set of matrices $D_j$, where each $D_j$ is defined as
$D_j = \lambda_j u_j u_j^H$,
$\lambda_j$ and $u_j$ are eigenvalues and eigenvectors, and $M$...
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0
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Deduce $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix}$ or the simple parity-check matrix just from the final matrix-vector product equations
A linear combination like $A_{1}+ A_{2}$ serves as a backup of $A_{1}$ when $A_{2}$ is known, and serves as a backup of $A_{2}$ when $A_{1}$ is known. As a result of linearity, any two out of $A_{1}x$,...
1
vote
0
answers
26
views
How to compute $RA$ in $O(n^2)$ operations instead of $O(n^3)$ using Householder reflection
I am currently writing a program that performs QR decomposition on a matrix $A$.
The guidelines to my assignment tell me that once I calculate $R$ using Householder reflections, there is a way to ...
0
votes
1
answer
38
views
Projection of vector
The projection of a vector $x$
onto a vector $u$ is
$proj_u(x) =\frac{\langle x, u \rangle}{\langle u, u \rangle}u.$
Projection onto $u$
is given by matrix multiplication
$proj_u(x)=Px$ where $P=\frac{...
user1290851
1
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0
answers
78
views
Symplectic approximation to a given matrix
I am interested in understanding methods to rigorously compute the closest symplectic matrix approximation to an arbitrary matrix $A$. A symplectic matrix $S$ satisfies the condition $ S^T J S = J $, ...
- 61
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0
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Interpretation of QR "values" (a la singular values)?
Let $A\in\mathbb{R}^{m\times n}$ with $m>n$ and $\text{rank}A=n$.
There exists $\hat{Q}\in\mathbb{R}^{m\times n}$ with orthonormal columns and $\hat{R}\in\mathbb{R}^{n\times n}$ upper triangular ...
