Orthogonalization
orthogonalization process
An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space an orthogonal system of non-zero vectors generating the same subspace in V . The most well-known is the Schmidt (or Gram–Schmidt) orthogonalization process, in which from a linear independent system a _ {1}, \dots, a _ {k} , an orthogonal system b _ {1}, \dots, b _ {k} is constructed such that every vector b _ {i} ( i = 1, \dots, k ) is linearly expressed in terms of a _ {1}, \dots, a _ {i} , i.e. b _ {i} = \sum _ {j= 1} ^ {i} \gamma _ {ij} a _ {j} , where C = \| \gamma _ {ij} \| is an upper-triangular matrix. It is possible to construct the system \{ b _ {i} \} such that it is orthonormal and such that the diagonal entries \gamma _ {ii} of C are positive; the system \{ b _ {i} \} and the matrix C are defined uniquely by these conditions.
The Gram–Schmidt process is as follows. Put b _ {1} = a _ {1} ; if the vectors b _ {1}, \dots, b _ {i} have already been constructed, then
b _ {i+ 1} = a _ {i+ 1} + \sum _ { j= 1} ^ { i } \alpha _ {j} b _ {j} ,
where
\alpha _ {j} = - \frac{( a _ {j+ 1} , b _ {j} ) }{( b _ {j} , b _ {j} ) } ,
j = 1, \dots, i , are obtained from the condition of orthogonality of the vector b _ {i+ 1} to b _ {1}, \dots, b _ {i} . The geometric sense of this process comprises the fact that at every step, the vector b _ {i+ 1} is perpendicular to the linear hull of a _ {1}, \dots, a _ {i} drawn to the end of the vector a _ {i+ 1} . The product of the lengths | b _ {1} | \dots | b _ {k} | is equal to the volume of the parallelepiped constructed on the vectors of the system \{ a _ {i} \} as edges. By normalizing the vectors b _ {i} , the required orthonormal system is obtained. An explicit expression of the vectors b _ {i} in terms of a _ {1}, \dots, a _ {k} is given by the formula
b _ {i} = \left | \begin{array}{cccc} ( a _ {1} , a _ {1} ) &\cdots &( a _ {1} , a _ {i- 1} ) &a _ {1} \\ \vdots &\ddots &\vdots & \vdots \\ ( a _ {i} , a _ {1} ) &\cdots &( a _ {i} , a _ {i- 1} ) &a _ {i} \\ \end{array} \right | (The determinant at the right-hand side has to be formally expanded by the last column). The corresponding orthonormal system takes the form q _ {i} = \frac{b _ {i} }{\sqrt {G _ {i- 1} G _ {i} } } , where G_i is the Gram determinant of the system a _ {1}, \dots, a _ {i} , with G0=1 by definition.
This process can also be used for a countable system of vectors.
The Gram–Schmidt process can be interpreted as expansion of a non-singular square matrix in the product of an orthogonal (or unitary, in the case of a Hermitian space) and an upper-triangular matrix with positive diagonal entries, this product being a particular example of an Iwasawa decomposition.
References
[1] | F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) |
[2] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |
Orthogonalization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonalization&oldid=52369