Singular Value Decomposition

Synonyms

SVD transformation; Latent semantic indexing; Principle component analysis

Definition

The SVD definition of a matrix is illustrated as follows [1]: For a real matrix $A = \left[ {a_{ij} } \right]_{m \times n}$, without loss of generality, suppose $m \ge n$ and there exists SVD of A (shown in Fig. 1):

$$A = U\left(\matrix{\Sigma _1 \cr 0}\right)V^T = U_{m \times m} \sum\nolimits_{m \times n} V_{n \times n}^{T}$$

where U and V are orthogonal matrices $U^T U = I_m$,$V^T V = I_n$. Matrices U and V can be respectively denoted as $U_{m \times m} = \left[ {u_1 ,u_2 , ..., u_m } \right]_{m \times m}$ and $V_{n \times n} = \left[ {v_1 ,v_2 , ..., v_n } \right]_{n \times n}$, where ${\rm{u}}_{\rm{i}} ,\left( {i = 1, ...,m} \right)$ is a m-dimensional vector $u_i = \left( {u_{1i} ,u_{2i} , ...,u_{mi} } \right)^T$ and $v_j ,\left( {j = 1, ...,n} \right)$ is a n-dimensional vector $v_j = \left( {v_{1j} ,v_{2j} , ..., v_{nj} } \right)^T$. Suppose rank(A) = r and...