All Questions
6 questions
1
vote
1
answer
1k
views
Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...
2
votes
0
answers
45
views
Combining SVD subspaces for low dimensional representations
Suppose we have matrix $A$ of size $N_t \times N_m$, containing $N_m$ measurements corrupted by some (e.g. Gaussian) noise. An SVD of this data $A = U_AS_A{V_A}^T$ can reveal the singular vectors $U_A$...
- 21
2
votes
1
answer
74
views
Limitation through the singular values
Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
- 143
1
vote
1
answer
737
views
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 $N$ ...
- 11
5
votes
1
answer
910
views
Proving a majorization inequality for the singular value of the product of two matrices without using tensor product
For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
- 115
2
votes
1
answer
375
views
Bound for matrix inner product based on singular values
Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
