Orthogonalization

orthogonalization process

An algorithm to construct for a given linear independent system of vectors in a Euclidean or Hermitian space $V$ 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)