Left Singular Vector

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



References

2013

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]