dvvtz: First derivative of the projection operator

Description

This function computes the first derivative of the projection operator $$P_V z= V V^\top z$$

Usage

dvvtz(v, z, dv, dz)

Arguments

orthonormal basis of the space on which z is projected. v is either a matrix or a vector.

vector that is projected onto the columns of v

first derivative of the the columns of v with respect to a vector y. If v is a matrix, dv is an array of dimension ncol(v)xnrow(v)xlength(y). If v is a vector,

first derivative of z with respect to a vector y. This is a matrix of dimension nrow(v)xlength(y).

Value

The first derivative of the projection operator with respect to y. This is a matrix of dimension nrow(v)xlength(y).

Details

For the computation of the first derivative, we assume that the columns of v are normalized and mutually orthogonal. (Note that the function will not return an error message if these assumptionsa are not fulfilled. If we denote the columns of v by $v_1,\ldots,v_l$, the first derivative of the projection operator is $$\frac{\partial P}{\partial y}=\sum_{j=1} ^ l \left[ \left(v_j z^ \top + v_j^ \top z I_n \right)\frac{\partial v_j}{\partial y} + v_j v_j ^ \top \frac{\partial z}{\partial y}\right]$$ Here, n denotes the length of the vectors $v_j$.

References