# 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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### Proof of the Single Value Decomposition

I am working through the a proof of the single-value decomposition, from Strang's 'Introduction to Linear Algebra, 4th edition'. I have included the proof as shown in the book at the end of this post. ...
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### Is here a mistake or can you explain me what orthonormal basis mean?

I am currently trying to get insight into SVD, and I found one book with an explanation of how we find the π and π matrices and why it holds that any πΓπ matrix can be represented in this ...
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### Properties of the eigenvalues of $A \cdot B$

Take two (real) matrices $A$ and $B$, where both have (real) eigenvalues within the unit circle, $B$ is also a diagonal matrix. Can I say something about the eigenvalues of the product $A \cdot B$ ? I ...
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### How do I normalize data before SVD?

Consider a satellite orbiting earth taking images. Since the problem can be approximated by the satellite moving with constant velocity and orientation relative to the fixed ground; the projection ...
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### Decomposition of a matrix into observability and controllability matrices

$\newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}$ I have a matrix $\boldsymbol{Q} \in \mathbb{R}^{M \times M}$ in ...
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### Singular value decomposation orthogonal to another matrix

I have two real matrices $A^{k\times m}$ and $B^{k\times n}$, let's assume $k\gg m$ and $m>n$. Let's also introduce an augmented matrix $C = [A \quad qB]$. I want to get the 'almost' singular ...
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### Discrepancies in Custom SVD Implementation Compared to np.linalg.svd - Sign Issues

I've been working on implementing a Singular Value Decomposition (SVD) algorithm from scratch in Python without using the np.linalg.svd function. My goal is to understand the underlying mechanics of ...
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### Solving A = RB using SVD

I have a linear system given by $\mathbf{A} = \mathbf{R}\mathbf{B}$ where $\mathbf{A}$ and $\mathbf{B}$ are 3-by-n matrices and $\mathbf{R}$ is a 3-by-3 orthonormal matrix (i.e., a rotation matrix). ...
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### Proof that if eigenvalues are equal to singular values then it is symmetric positive-semi definite

Take $A\in R^{kxk}$. Suppose that its eigenvalues are equal to its singular values. Then show that $A$ is symmetric and positive semi-definite. I've found sources stating it but I haven't managed to ...
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### Deriving the SVD from the eigendecomposition

If $A$ is a rectangular matrix of dimensions $m\times n$, then $S_L=AA^T$ and $S_R=A^TA$ are square symmetric matrices. Hence, using the eigendecompostion we can write $$S_L=AA^T=U\Lambda_{S_L} U^T$$...
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### What to do when the Gram matrix of an underdetermined system is singular?

I am currently trying to speedrun my linear analysis course, I have been doing pretty well so far, but hit a wall when the lectures started hitting on SVD and under/overdetermined systems of equations....
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### Geometric interpretation of left-singular and right-singular vectors

I wanted to ask if and how $A^{T}A$ respectively $AA^{T}$ can be interpreted geometrically in the sense of it's eigenvectors being the left and right singular vectors? What is the geometric ...
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### Most General Solution of a Matrix Equation (Arising From SVD)

Suppose we have an arbitrary but known $n\times m$ complex matrix $A\in\textbf C^{n\times m}$ which therefore has an $m\times n$ conjugate transpose $A^{\dagger}\in\mathbf C^{m\times n}$. Now suppose ...
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### Confusion regarding the geometrical meaning of singular values in SVD

I am trying to visualize in MATLAB the relationship between the singular value decomposition (SVD) of a matrix of points. To simplify the problem, I am working in 2D and I am considering an ellipse ...
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### Efficient SVD of low-rank matrix of the form $C=AB^{T}$
Let $A,B$ be two real matrices, of dimensions $n \times k$ and $m \times k$, respectively. I assume that $k \ll n,m$. I am interested in computing the SVD of the product matrix $C = AB^{T}$. The ...