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Orthogonal or Unitary Matrices

 

A real orthogonal or complex unitary matrix (usually denoted Q) is often represented  in ScaLAPACK as a product of elementary reflectors -- also referred to as     elementary Householder matrices (usually denoted tex2html_wrap_inline14421). For example,
displaymath14415
Most users need not be aware of the details, because ScaLAPACK routines are provided to work with this representation:

The following details may occasionally be useful.

An elementary reflector (or elementary Householder matrix) H of order n is a unitary matrix  of the form    
 equation2204
where tex2html_wrap_inline14435 is a scalar and v is an n-vector, with tex2html_wrap_inline14441); 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.

Some redundancy in the representation (3.4) exists, which can be removed in various ways. Like LAPACK, the representation used in ScaLAPACK (which differs from that used in LINPACK or EISPACK) sets tex2html_wrap_inline14449; hence tex2html_wrap_inline14451 need not be stored. In real arithmetic, tex2html_wrap_inline14453, except that tex2html_wrap_inline14455 implies H = I.

In complex arithmetic , tex2html_wrap_inline14435 may be complex and satisfies tex2html_wrap_inline14461 and tex2html_wrap_inline14463. 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 tex2html_wrap_inline14435 to be complex is that, given an arbitrary complex vector x, H can be computed so that
displaymath14416
with real tex2html_wrap_inline14473. 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 [94].


next up previous contents index
Next: Algorithmic Differences between LAPACK Up: Contents of ScaLAPACK Previous: Generalized Symmetric Definite Eigenproblems

Susan Blackford
Tue May 13 09:21:01 EDT 1997