# 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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### Is there a name for the matrices with the same eigenvectors?

I am given a symmetric positive semi-definite matrix $A\in\mathbb{R}^{n\times n}$. I perform the SVD, $$A = V \Sigma V^\top,$$ where $V$ is unitary and $\Sigma$ is diagonal. Next, I pick up another ...
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### Determining A Vector Through the Center of Multiple Points on a Sphere

I am working on a machine vision task that requires me to determine spin rate and spin axis of a moving ball. I have had some luck, and actually do have a solution but am looking for a more efficient ...
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### Why are Latent Factor Models called "SVD"?

Latent factor models (usually used for recommender systems) are a matrix decomposition of matrix $R$ such that $$R = P \cdot I^T$$ with the "twist" that values in $R$ can be missing (which ...
1 vote
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### Polar decomposition of a linear combination of unitary matrices

Consider a complex-valued square matrix $M$ of the form $$M = \frac{1}{2}\left(U_1 + e^{-i\phi}U_2\right),$$ where $U_1$ and $U_2$ are unitary matrices and $\phi$ is a real number. Moreover, consider ...
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### Moore-Penrose pseudoinverse solves the least squares problem (SVD framework) [duplicate]

I am a computer science researcher who has to learn some numerical linear algebra for my work. I have been struggling with the SVD and Moore-Penrose pseudoinverse as of late. I am trying to solve some ...
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### Matrix as a dataset vs a transformation

In my machine learning class, we are learning about Singular Value Decomposition (SVD) on a data matrix. SVD allows us to "decompose" a $nxm$ matrix, $A$, into a product of three "...
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### Upper bound on normwise relative error

Suppose I have two vectors $y_1, y_2 \in \mathbb{R}^{n\times 1}$ which are both transformations of the same vector $x$ by two different matrices $T_1, T_2 \in \mathbb{R}^{m\times n}$, i.e.: \begin{...
1 vote
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### $\exists L>0, \forall m,n\in\mathbb{N}, \forall A,B \in \mathbb{R}^{n\times m}, \|\sqrt{AA^T}-\sqrt{BB^T}\| \le L \|A-B\|$?

Is it true that \begin{equation} \exists L>0, \forall m,n\in\mathbb{N}, \forall A,B \in \mathbb{R}^{n\times m}, \|\sqrt{AA^T}-\sqrt{BB^T}\| \le L \|A-B\|, \end{equation} where $\|\cdot\|$ is the ...
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### Is there a formula for the norm of an orthogonal projection?

In all introductory linear algebra texts there is a discussion on orthogonal projection. Let $u = w_1 + w_2$, where $w_1$ is the projection of $u$ along $v$ and $w_2$ is projection of orthogonal to $v$...
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### Why does this algorithm for the Takagi factorization fail here?

In this paper it was found that the Takagi factorization of a complex symmetric matrix $A$ can be calculated from the singular value decomposition $A=U\Lambda V^H$ by $$A = X\Lambda X^T \tag{1}$$ ...
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### SVD of an orthogonal projector

Here is my observation: Suppose there is an orthogonal projector $P$ such that $P=P^2$. Then for arbitrary $x$, $Px$ and $(I-P)x$ are orthogonal. So we have $$x^* P^* (I-P)x=0$$ where $A^*$ means ...
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### How does the SVD behave under a "phase transformation"?

The singular value decomposition of a complex matrix $A$ can be written as $$A= UDV^{H}$$ where $U$ and $V$ are hermitian matrices and $D$ is a diagonal matrix with entries $(D)_{ii}=\sigma_i\geq0$ ...
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### Some questions regarding svd computations

0 I've some questions regarding svd: Consider a matrix A ∈ R500×5 and its SVD [U, S, V T ] = svd(A). (Assume A is centered). a) is the second left singular vector of A is the direction in R5 with the ...
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### Singular value decomposition for a convolution kernel

I am trying to get some insight into how to solve the following linear equations: $$\frac{dA_n}{dt} = i\sum\limits_{m} B^*_{n-m} C_m$$ $$\frac{dC_n}{dt} = i\sum\limits_{m} B_{n-m} A_m$$ where $B_n$ ...
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### Singular Value Decomposition (SVD) with Monotonic Constraint

I am trying to compress some cumulative distribution functions (CDFs) which are stored in an $N \times M$ matrix $A$. Each of the $M$ columns contains $N$ monotonically-increasing values which might ...
1 vote
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### What does "up to complex signs" mean?

I'm studying the SVD, but I'm confused about the term "up to complex signs" If A is square and the singular values are distinct, the left and right singular vectors are uniquely determined ...
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### Singular value decomposition for linear transformations (operators)

I find that the singular value decomposition expression is often written for a matrix instead of a linear transformation. Accordingly I have the following (fairly basic) question on my mind. If $A$ ...
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### Matrix inverse step in SVD & ridge regression

When we do OLS of $y$ on $X$, with $X$ being a n x p input matrix, the OLS $\beta$ is $(X^TX)^{-1}X^TY$, and the Ridge regression beta is $(X^TX+\lambda I)^{-1}X^TY$. Also, the singular value ...
1 vote
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### SVD and eigenvectors

Let $A = U \Sigma V^{H}$ be the SVD. I must show that columns $v^{1}, \dots, v^{n} \in V$ form a complete set of eigenvectors of $AA^{H}$. Note that $\text{rank}(A) = r$ and $m \ge n \ge r$. I simply ...
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### Estimating the right singular vector corresponding to an all positive left singular vector.

Suppose that $A$ is a $k\times n$ real matrix (and for intuition suppose that $k$ is very large) that has SVD $U\Sigma V^\dagger$. Suppose further that we know that there is exactly one column of $U$ ...
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### SVD of residual matrix non-orthogonal to orthogonal projection?

Suppose we have two data matrices, $X$ which is a $m$ x $n$ matrix, where $m$ >> $n$ $Y$ which is a $n$ x $p$ matrix, consisting of $p$ orthonormal columns, where $n$ > $p$ Next, we find ...
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### Why should unit vectors that get linearly mapped to the semi-axes of an ellipse be orthogonal?

I'm reading Wikipedia's visual proof of the singular value decomposition. They say: To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces –...
Given a Hermitian matrix $C^{n \times n}$ with a known spectral decomposition $U \Delta U^{-1}$. Is there any way to do a low-rank approximation of $H$ without computing the SVD of $H$ from scratch?
Let $F(A)$ be a matrix-valued function, operating on real-valued matrix $A \in \mathbb{R}^{m, n}$ that applies a scalar function $f(\lambda)$ on the singular values of $A$. That is, suppose $A$ has ...