# Questions tagged [svd]

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.

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### Does Sharing same singular vectors lead to some properties?

What properties do a pair of matrices have if they share same ordered right singular vectors? Or for a specical case the matrices are square, what properties do they have if their ordered eigen-...
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### Gradient of $A \mapsto \sigma_i (A)$

Let $A$ be an $m \times n$ matrix of rank $k \le \min(m,n)$. Then we decompose $A = USV^T$, where: $U$ is $m \times k$ is a semi-orthogonal matrix. $S$ is $k \times k$ diagonal matrix , of ...
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### Stability of eigenvalues/singular values on altering the matrix

From Strang's Introduction to Linear Algebra (p375), there is a paragraph on the instability of eigenvalues in relation to the stability of singular values when $A$ is altered slightly. Moreover, it ...
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### Simple estimator for biggest singular value

I was reading a paper where they used $\sqrt{\|A\|_\infty \|A\|_1 }$ as an estimate of the biggest singular value $\sigma_{max}(A)$ of a non-singular matrix $A \in \mathbb{R}^{n \times n}$. However, ...
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### SVD in multilinear algebra

I'm completely new in this field of algebra and I would like to better understand the extension of the SVD in the multilinear case. In particular, I'm interested in understanding what the so-called "...
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### SVD uniquely determined for a matrix

I'm trying to prove the following: Given a matrix, the singular values of that matrix are uniquely determined. If that matrix is square, and the singular values are distinct, the left and right ...
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### How does the outer product work in matrix approximation using SVD?

SVD factorizes a matrix $A \in \mathbb{R}^{mxn}$ such that $A = U\Sigma V^T$, quoting from Deisenroth et al: Instead of doing the full SVD factorization, we will now investigate how the SVD allows ...
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### Every matrix is diagonal

Using SVD, prove the following: Let $A \in \mathbb{R}^{m\times n}$ and $T(x)=Ax$, a linear transformation for which A is the representative matrix in the canonical coordinates. The existence of SVD ...
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### Advantage of singular value decomposition of $X$ over eigenvalue analysis of $XX^T$

Suppose I have a real $n \times N$ matrix $X$ where $n$ is a small number like $<20$ and $N$ is a big number like $>100000$. For example $X$ might be a time series of $n$ measurment quantities ...
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### Diagonal matrices are not necessarily square?

I have seen it implied, in the context of singular value decomposition, that diagonal matrices are not necessarily square. Is this true? How can it be true? Can someone please explain this in more ...
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### Pseudoinverse of $\mathbf{A} \in \mathbb{R}^{m \times n}$ multiplied by $\mathbf{A}$

The Moore-Penrose pseudoinverse of a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ is \begin{equation} \mathbf{A}^+ = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T. \end{equation} Now, using this we ...
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### Independence of singular values

If $X$ is a matrix valued random variables, are its singular values independent as random variables?
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### Classification with SVD - Eigenfaces

I just saw a play list on Youtube where Professor Brunton teach how SVD works and its applications. He mention that with SVD, classification can be done. https://www.youtube.com/watch?v=gXbThCXjZFM I ...
Consider the SVD decomposition of a data matrix $$X = UDV^T$$ where rows of $X$ are samples. In Elements of Statistical Learning it is said that Then $Z = UD$ is the matrix of principal ...