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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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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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
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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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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
$$
A =\begin{bmatrix}
2& 9& 2& 1& 4& 1& 0& 0& 0 \\
9& 65& 9& 1& 4& 1& 0& 0&...
- 1,261
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Compressed image using SVD draws a clear line between part that's blank and part with a drawing. Why? [closed]
I'm trying to compress grayscale images using SVD. This is the original image:
Yes, there's a lot of blank space.
I then choose the x% largest singular values, perform the transformed matrices ...
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1
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For a real symmetric matrix, is the product of two factors of its rank decomposition (right times left) also symmetric?
Recently, I am learning generalized inverse of a matrix. Given a real symmetric matrix $A\in\mathbb{R}^{n\times n}$ with ${\rm rank}(A)=r$, suppose the rank decomposition of $A$ is given as follows:
$$...
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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
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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 ...
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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 ...