Right-singular vector

Figure 4.5 The two significant right singular vectors obtained after SVD of the matrix A. Note that all vectors in this figure have maximal absorbance of the order of 1 AU. Y Pan, L. Susithra,...

Figure 4.8 The first six right singular vectors of the matrix (1) first vector (2) second vector (3) third vector (4) fourth vector (5) fifth vector (6) sixth vector. (C. Li, E. Widjaja,...

The quantities of are called the singular values of A and the columns of U and V are called the left and right singular vectors. If A is symmetric, positive-semidefinite, the eigenvalues and the singular values of A are equal if A is not symmetric, they are not. 

U has dimensions pJ x a and contains the a left eigenvectors of Hjk- S is a X a and contains the singular values (SV). V is Kp x a and contains the a right singular vectors of the decomposition. The SVD matrices in Eq. 4.65 include only the SVs and singular vectors corresponding to the a state variables retained in the model. The full SV matrix S has dimension... 

For a situation with one dependent variable, a score vector is sought which has maximal covariance with y. For the two-way case the loading vector is found directly as X y/HX yH. For the three-way case the weights can also be found directly as the first left and right singular vectors from a singular value decomposition of a matrix Z (J x K) which is obtained as the inner product of X and y (see proof in Appendix 6.B). Each element in Z is given by... 

Determine wd and wj as the first left and right singular vectors of Z. 

Let U be the first R left singular vectors of [Xi X2] and V the first R right singular vectors of [XjIXj]. 

Compute the weights along the second and third modes for the component, y/f and y/f, as the first left and right singular vectors of Z. 

The right singular vectors vl l,. .., are orthonormal. Therefore, any vector jc e can be written as the Unear combination... 

Let us now define a second vector y e St that is a linear combination solely of the right singular vectors for the zero singular values. 

Written in terms of the left and right singular vectors, the solution is... 

The advantage of the SVD approach becomes evident when we do not have sufficient data to determine all coefficients Pj. Then, the right singular vectors v aj = 0 e 91 provide information about the missing data points x that are necessary to determine all Pj. In particular, we should add new measurements. .., such... 

We see from S that the third singular value is zero, and identify the missing data from the corresponding right singular vector... 

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