Representation of Orthogonal or Unitary Matrices

A real orthogonal or complex unitary matrix (usually denoted Q) is often represented in LAPACK as a product of elementary reflectors -- also referred to as elementary Householder matrices (usually denoted H_i). For example,

$$Q = H_{1} H_{2} \ldots H_{k}.$$

Most users need not be aware of the details, because LAPACK routines are provided to work with this representation:

• routines whose names begin SORG- (real) or CUNG- (complex) can generate all or part of Q explicitly;

• routines whose name begin SORM- (real) or CUNM- (complex) can multiply a given matrix by Q or Q^H without forming Q explicitly.

The following further details may occasionally be useful.

An elementary reflector (or elementary Householder matrix) H of order n is a unitary matrix of the form

$$H = I - \tau v v^{H}$$ (5.1)

where $\tau$ is a scalar, and v is an n-vector, with $\vert \tau \vert ^2 \vert\vert v \vert\vert _2 ^2 = 2 \rm {Re}(\tau$); v is often referred to as the Householder vector . Often v has several leading or trailing zero elements, but for the purpose of this discussion assume that H has no such special structure.

There is some redundancy in the representation (5.1), which can be removed in various ways. The representation used in LAPACK (which differs from those used in LINPACK or EISPACK) sets v₁ = 1; hence v₁ need not be stored. In real arithmetic, $1 \leq \tau \leq 2$, except that $\tau = 0$ implies H = I.

In complex arithmetic, $\tau$ may be complex, and satisfies $1 \leq \rm {Re}(\tau) \leq 2$ and $\vert \tau - 1 \vert \leq 1$. Thus a complex H is not Hermitian (as it is in other representations), but it is unitary, which is the important property. The advantage of allowing $\tau$ to be complex is that, given an arbitrary complex vector x, H can be computed so that

$$H^H x = \beta (1, 0, \ldots , 0)^T$$

with real $\beta$. This is useful, for example, when reducing a complex Hermitian matrix to real symmetric tridiagonal form, or a complex rectangular matrix to real bidiagonal form.

For further details, see Lehoucq [79].