Concept

Householder operator

Summary
In linear algebra, the Householder operator is defined as follows. Let V, be a finite-dimensional inner product space with inner product \langle \cdot, \cdot \rangle and unit vector u\in V. Then : H_u : V \to V, is defined by : H_u(x) = x - 2,\langle x,u \rangle,u,. This operator reflects the vector x across a plane given by the normal vector u. It is also common to choose a non-unit vector q \in V, and normalize it directly in the Householder operator's expression: :H_q \left ( x \right ) = x - 2, \frac{\langle x, q \rangle}{\langle q, q \rangle}, q ,. Properties The Householder operator satisfies the following properties:
  • It is linear; if V is a vector space over a field K, then :\forall \left ( \lambda, \mu \right ) \in K^2, , \forall \left ( x, y \right ) \in V^2, , H_q \left ( \lambda x + \mu y \righ
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