High Performance Programming for Soft Computing

316 High Performance Programming for Soft Computing

3. The orthogonal vector should be normalized,

(15.3.24)

4. In the j

th

step, the vector (15.3.25) is found,

(15.3.25)

Therefore, the Gram-Schmidt process allows us to construct the

orthonormal set which spans a k-dimensional subspace

in

.

For arbitrary , and , the linear operator A provides a

map if (15.3.26) is satisfi ed.

121 2

(| |) | | ,Acxc cAxcAy+= +

(15.3.26)

For the orthonormal basis and the arbitrary vector in ,

where , using the linear operator implies that,

(15.3.27)

The Hermitian conjugate of the linear operator for the

arbitrary vectors

, is defi ned