Left Singular Vector

A Left Singular Vector is a matrix vector from the left orthogonal matrix in a singular value decomposition structure.

• • …
• Counter-Example(s):
  • a Right Singular Vector.
• See: Eigenvector, Singular Value, Singular Value Decomposition, 2nd Left Singular Vector.

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References

• http://wikipedia.org/wiki/Singular_value_decomposition#Singular_values.2C_singular_vectors.2C_and_their_relation_to_the_SVD

2013

• (Wikipedia, 2013) http://en.wikipedia.org/wiki/Singular_value_decomposition
  • … Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form :[math]\displaystyle{ M = U\Sigma V^*, \, }[/math] where U is a m×m real or complex unitary matrix, Σ is an m×n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and V* (the conjugate transpose of V) is an n×n real or complex unitary matrix. The diagonal entries Σ_i,i of Σ are known as the singular values of M. The m columns of U and the n columns of V are called the left-singular vectors and right-singular vectors of M, respectively.

    The singular value decomposition and the eigendecomposition are closely related. Namely:

    • The left-singular vectors of M are eigenvectors of MM*.
    • The right-singular vectors of M are eigenvectors of M*M.
    • …

2012

• (Golub & Van Loan, 2012) ⇒ Gene H. Golub, and Charles F. Van Loan. (2012). “Matrix Computations (4th Ed.)." Johns Hopkins University Press. ISBN:1421408597
  • QUOTE: If [math]\displaystyle{ A }[/math] is a real m-by-n matrix, then there exist orthogonal matrices :[math]\displaystyle{ U = \bigl [ u_1,...,u_m \bigr] \in \mathbb{R}^{m \times m} \ \text{ and } \ V = \bigl [ v_1,...,v_n \bigr] \in \mathbb{R}^{n \times n} }[/math] such that :[math]\displaystyle{ U^TAV = \Sigma = \operatorname{diag}(\sigma_1,...,\sigma_p) \in \mathbb{R}^{m \times n} \ = \ \operatorname{min}\{m,n\} }[/math] where [math]\displaystyle{ σ_1 ≥ σ_2 ≥ … ≥ σ_p ≥ 0 }[/math]. …

    … The [math]\displaystyle{ σ_i }[/math] are the singular values of [math]\displaystyle{ A }[/math], the [math]\displaystyle{ u_i }[/math] are the left singular vectors of [math]\displaystyle{ A }[/math], and the [math]\displaystyle{ v_i }[/math] are right singular vectors of [math]\displaystyle{ A }[/math]. Separate visualizations of the SVD are required depending upon whether [math]\displaystyle{ A }[/math] has more rows or columns. Here are the 3-by-2 and 2-by-3 examples: :[math]\displaystyle{ \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ u_{21} & u_{22} & u_{23} \\ u_{31} & u_{32} & u_{33} \end{bmatrix}^T \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} \begin{bmatrix} v_{11} & v_{12} \\ v_{21} & v_{22} \end{bmatrix} = \begin{bmatrix} \sigma_{1} & 0 \\ 0 & \sigma_{2} \\ 0 & 0 \end{bmatrix}. }[/math] :[math]\displaystyle{ \begin{bmatrix} u_{11} & u_{12} \\ u_{21} & u_{22} \end{bmatrix}^T \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \begin{bmatrix} v_{11} & v_{12} & v_{13} \\ v_{21} & v_{22} & v_{23} \\ v_{31} & v_{32} & v_{33} \end{bmatrix} = \begin{bmatrix} \sigma_{1} & 0 & 0 \\ 0 & \sigma_{2} & 0 \end{bmatrix} . }[/math]

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