All Questions
Tagged with matrix-decomposition svd
293
questions
0
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2
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51
views
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
0
votes
0
answers
27
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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
1
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answers
23
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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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0
answers
25
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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....
- 115
4
votes
1
answer
90
views
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
1
vote
0
answers
28
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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 ...
0
votes
1
answer
33
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Distance between subspaces with spectral norm
I was trying to prove this following theorem ,
Let
$$ W=\begin{bmatrix}
\underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2}
\end{bmatrix} $$
$...
0
votes
0
answers
24
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Is polar decomposition commutative for diagonal matrices?
I did a error while understanding about the polar decompositon.
I thought polar decomposition is PU, but it is UP. While trying ...
- 111
-1
votes
1
answer
34
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Decomposing a matrix with unit sphere constraints [closed]
I would like to decompose an $m\times n$ matrix $A$ into two matrices $U\in\mathbb{R}^{m\times n}$ and $V\in\mathbb{R}^{n\times n}$ such that $UV=A$, and the $m$ rows of $U$ each have unit magnitude. ...
0
votes
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answers
37
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Reasons of computing smallest eigenvalue $R^TR$ instead of singular value
I have problems in understanding why author of this article uses smallest eigenvalue of a cross product matrix instead of a data matrix. I know that $SVD(AA^T)=UD^2U^T$, but I don't know why not ...
- 1
2
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0
answers
47
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Is there an "orthogonal factorization" of bivariate functions that is analogous to the SVD of matrices?
For a matrix $X \in \mathbb{R}^{m\times n}$, we have the SVD decomposition
$$ X = U D V^\top, $$
where $U\in\mathbb{R}^{m\times r},\ V\in\mathbb{R}^{n\times r}$ are orthonormal matrices and $D=\text{...
- 157
1
vote
1
answer
33
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SVD decomposition of a square matrix of complex numbers
Le $M$ be any matrix in $C^{n \times n}$. Consider the matrix $MM^*$.
This matrix is Hermitian ($(MM^*)^* = MM^*$), and positive semi-definite
($\forall v^*, v^*MM^* v = (v^* M) (M^* v) = (v^*M) (v^*M)...
- 19
1
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0
answers
49
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Decomposition of a matrix into observability and controllability matrices
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I have a matrix $\boldsymbol{Q} \in \mathbb{R}^{M \times M}$ in ...
- 57
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29
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Phase degeneracy of singular vectors in SVD of complex matrix
An $m*n$ complex matrix $M$ is given. I take its SVD, $M=U\Sigma V^\dagger$. My question is, are the singular vectors $\textbf{u}_n$ and $\textbf{v}_n$ degenerate? Can I pick any phase for one that ...
- 21
3
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0
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98
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Weighted Nearest Kronecker Product
For a given $m n$-length vector $a$, the problem of finding an $m$-length vector $x$ and an $n$-length vector $y$ that minimize
$$
\lVert a - x \otimes y \rVert^2
$$
is known as the Nearest Kronecker ...
