Mathematics Stack Exchange
Mathematics Stack Exchange

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.

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix.

Formally, the singular value decomposition of an $m \times n$ real or complex matrix $M$ is a factorisation of the form $UAV^*$ where $U$ is an $m\times m$ real or complex unitary matrix, $A$ is an $m\times n$ rectangular diagonal matrix with non-negative real numbers on the diagonal, and $V$ is an $n\times n$ real or complex unitary matrix.

The singular-value decomposition can be computed using the following observations:

  • The left-singular vectors of $M$ are a set of orthonormal eigenvectors of $MM^*$.
  • The right-singular vectors of $M$ are a set of orthonormal eigenvectors of $M^*M$.
  • The non-zero singular values of $M$ (found on the diagonal entries of $A$) are the square roots of the non-zero eigenvalues of both $M^*M$ and $MM^*$.

Source: Wikipedia.

1502 questions
10
votes
1 answer

Plane fitting using SVD normal vector

I am trying to fit the plane using SVD. The algorithm that I am using is following: From each point subtract centroid Calculate SVD Find normal as 3rd column of matrix $U$ I have 7 points to fit. So matrix $A$ is $7\times3$. Matrix U is…
Tomas
  • 103
  • 1
  • 6
7
votes
1 answer

Are reduced SVD and truncated SVD the same thing?

Truncated SVD: http://scikit-learn.org/stable/modules/generated/sklearn.decomposition.TruncatedSVD.html Reduced SVD, I thought this is essentially the same thing, and it appears to be actually more commonly called this way. If you could provide…
zyxue
  • 555
6
votes
3 answers

Understanding a derivation of the SVD

Here's an attempt to motivate the SVD. Let $A \in \mathbb R^{m \times n}$. It's natural to ask, in what direction does $A$ have the most "impact". In other words, for which unit vector $v$ is $\| A v \|_2$ the largest? Denote this unit vector as…
eternalGoldenBraid
  • 2,089
4
votes
1 answer

Merging two SVD factorizations but using only $U_1S_1$ and $U_2S_2$ (question about Radim's Phd Thesis [gensim])

I am reading Radim's Phd Thesis (the creator of gensim), on the chapter about SVD (page 45). There he puts certain assignment, indicating that we can calculate the "merge" of a couple of SVD factorizations $A_1=U_1S_1V_1^{T}$ and…
user251053
  • 51
  • 3
3
votes
2 answers

how SVD is calculated in reality

let us suppose that we have following matrix $ A= \left[ {\begin{array}{cc} 4 & 0 \\ 3 & -5 \\ \end{array} } \right] $ for calculation of SVD,first i have calculated $A'*A$ which is equal to $ A'*A= …
dato datuashvili
  • 9,194
3
votes
2 answers

Number of Singular Values

Is there any equation which describes or estimates the number of singular values of a Matrix $X$ ? I found out that the number is equal to the number of eigenvalues of the Matrix $X^{*} X$, which are calculates as: $det( \lambda *I- X^{*} X)=0$.…
N8_Coder
  • 61
  • 1
  • 7
3
votes
1 answer

When will eigenvalues of a symmetric matrix repeat?

When we calculate singular values in Singular value decomposition we use the common eigenvalues (positive square roots) of $A^TA$ or $AA^T$, where $A$ is an $m\times n$ real matrix. We know that singular values of A may be repeated. Now I am trying…
P Pradhan
  • 77
2
votes
1 answer

How singular is a matrix?

Any $m\times n$ matrix has a singular value decomposition, $$A = USV^T\text{,}$$ where $A$ is $m\times n$ matrix, and its singular values are $(\sigma_1,...,\sigma_m)$. There is a method to judge how singular $A$ is by computing the ratio of the…
avocado
  • 1,209
2
votes
1 answer

SVD and frobenius norm

If I have an SVD, $A = X \Sigma Y$, where $\Sigma \in \mathbb{R}^{m \times n}$ has the singular values on the diagonal and $X \in \mathbb{R}^{m \times m}$ and $Y \in \mathbb{R}^{n \times n}$ are orthogonal matrices, then does the squared frobenius…
User_13
  • 111
2
votes
1 answer

SVD [Singular Value Decomposition] on Transformation Matrix

svd(T) = u sigma v Here I understand meaning of each and every term and why SVD is important. But I am failing to interpret this equation from Linear Algebra glasses. When I have learnt about Linear Algebra, there was one thing common in all…
LLB
  • 29
2
votes
2 answers

SVD when $A-\lambda I = 0$.

Leting $$A=\left[\begin{array}{rrrr}-1&1&1&1\\1&-1&1&1\\1&1&-1&1\\1&1&1&-1\end{array}\right],$$ find SVD, i.e., $A=U\Sigma V^T$ with $U^TU=UU^T=I$ and $V^TV=VV^T=I$, where $I$ is identity matrix. Then, since $$A^TA = AA^T=4I,$$ I could found…
Danny_Kim
  • 3,423
2
votes
2 answers

SVD -obligation of normalization

let us suppose we have following matrix $ A= \left[ {\begin{array}{cc} 2 & 2 \\ -1 & 1 \\ \end{array} } \right] $ and i want to compute SVD of this matrix, i have calculated first of all $A*A'$ , which is…
dato datuashvili
  • 9,194
2
votes
1 answer

interpretation of SVD for text mining topic analysis

Background I'm learning about text mining by building my own text mining toolkit from scratch - the best way to learn! SVD The Singular Value Decomposition is often cited as a good way to: Visualise high dimensional data (word-document matrix) in…
MYO TextMiningToolkit
  • 21
2
votes
2 answers

SVD of a matrix from SVD of its columns

Assume a matrix A, and I know the left singular vectors of SVD(A(:,i)), i=1,2,...,# of columns, is there a simple/fast transformation to obtain the left singular vectors of SVD(A) (the whole matrix)?
Gezheng
  • 21
1
vote
0 answers

When QR outperform SVD to estimate column space of a symmetric matrix?

I have a symmetric matrix, $\hat{A}, $which is an estimator of another true symmetric matrix, $A$. I would like to estimate the column space of $A$. I know I can do QR-Algorithm or use spectral decomposition to find non-zero eigenvectors of…
bankrip
  • 566
1
2 3 4