All Questions
Tagged with matrix-decomposition singular-values
52
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Solving Matrix Equation using SVD
I'm reading this paper by Bishop and Tipping.
They solve the equation
$$(SC^{-1} - I)W = 0$$
Where $W \in \mathbb{R}^{d \times q}$ and $S , C \in \mathbb{R}^{d \times d}$ and $C = WW^T + \sigma^2 I$ ...
- 43
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102
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Scale Invariant Singular Value Decomposition
I am looking for a reference to the concept of Scale Invariant SVD which is mentioned in the "variations and generalisations" section of the Wikipedia article for SVD:
https://en.wikipedia....
- 41
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Understanding the proof of the sum of eigenvalues and singular values
I am trying to Understand this proof . So far I understand everything but the part where the author says the following:
"Due to Schur decomposition, there exist a unitary matrix $U$ and an upper ...
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1
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SVD of product of diagonal and unitary matrices
Given two (possibly rectangular) diagonal matrices $\Sigma_\text{L}$ and $\Sigma_\text{R}$ with nonnegative elements, what can we say about the singular value decomposition of
$$\Sigma_\text{L} X \...
- 73
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1
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Singular value of identity minus 1-rank matrix
Given vectors $u,v\in \mathbb{R}^d$, I wonder what we can say about the minimum singular value of $I-uv^\top$? I know that when $u=v$, this matrix is symmetric so it is not hard to compute this. ...
- 115
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535
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Why the singular values in SVD are always hierarchical/descending?
Please, I'm trying to understand why singular values (SV) are always hierarchical/descending. At the beginning of my studies, I thought that the hierarchy of sigmas ($ \sigma_1 \geq \sigma_2 \geq ... \...
- 19
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1
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69
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Are the following statements true regarding the singular values of a real matrix?
Let $A\in \mathbb{R}^{m\times k}\ (k<m)$ be a real matrix. Suppose, $\forall x \in \mathbb{R}^{k}$, and for a $\delta \in (0,1)$, the following inequality holds:
\begin{equation}
(1-\delta) \|x\|...
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1
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Given a symmetric matrix $H$, can we prove $x^THx\le \sigma(H)\|x\|_2^2$?
Given a symmetric matrix $H$, can we prove that $$x^THx\le \sigma(H)\|x\|_2^2$$ where $ \sigma(H)$ is the maximal singular value of $H$?
- 2,883
2
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1
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118
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General form of the null space of an orthogonal projection operator
I have a $2n\times n$ real matrix $A$ which has full rank $n$. I would like to zero value eigenvectors of $P=AA^+$, i.e. the orthogonal projector onto the range of $A$.
$A^+$ is the Moore-Penrose ...
- 429
4
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903
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Does a singular value decomposition always exist for complex matrix?
I know for real matrix A, SVD always exists, but I am wondering for any complex matrix, will SVD still exist for any scenarios? Thanks.
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Why should $\arg\max_{\Vert x \Vert = 1} \Vert A x\Vert$ be a linear combination of the rows of $A$?
From the singular value decomposition of an $m \times n$ matrix $A$, we should have $\Vert A \Vert = \sigma_1$, where $\sigma_1$ is the top singular value of $A$.
This would mean that the maximum ...
- 1,332
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Limitation through singular value of a matrix
Question: Given a matrix $X \in \mathbb{R}^{m \times n}$. Let $(X^k)_k \in \mathbb{R}^{m \times n}$ is a sequence converge to $X$. Denote $\sigma_i(X)$ as singular values of $X$. Prove that $$\lim_{k \...
- 873
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How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?
Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...
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When do Linear Transformations NOT preserve angles between vectors? Doesn't the SVD tell us all linear transformations preserve angles?
From searching on the internet, I learned only a subset of linear transformations preserve angles between vectors. But -
Learning about the SVD - we can geometrically understand as breaking down some ...
- 887
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In singular value decomposition, is there a difference between starting with transpose(A) * A or A * transpose(A)?
When calculating the SVD for a matrix I go through these steps :
1/ Calculate tranpose(A)*A
2/ Find the eigenvalues for that matrix and deduce $\Sigma$
3/ Find the eigenvectors and deduce $V$
4/ ...
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