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Concept
Householder operator
Summary
In linear algebra, the Householder operator is defined as follows. Let be a finite-dimensional inner product space with inner product and unit vector . Then
is defined by
This operator reflects the vector across a plane given by the normal vector .
It is also common to choose a non-unit vector , and normalize it directly in the Householder operator's expression:
The Householder operator satisfies the following properties:
It is linear; if is a vector space over a field , then
It is self-adjoint.
If , then it is orthogonal; otherwise, if , then it is unitary.
Over a real or complex vector space, the Householder operator is also known as the Householder transformation.
Official source
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