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Re: Given a matrix A,

• To: mathgroup at smc.vnet.net
• Subject: [mg83370] Re: [mg83326] Given a matrix A,
• From: danl at wolfram.com
• Date: Sun, 18 Nov 2007 04:54:49 -0500 (EST)
• References: <200711171030.FAA09684@smc.vnet.net>

> Given a matrix A with (m, n) dimension and m<n, I want to construct a
> semi-orthogonal matrix L with dimension (n, n-m) [that is, L'L = I,
> where I denotes an identity matrix with dimension (n-m)] such that AL
> = 0.

orthogonalComplement[mat_?MatrixQ] :=
  Transpose[Orthogonalize[NullSpace[mat]]]

Example:

In[49]:= mat = RandomInteger[{-10, 10}, {3, 7}];
ocmat = orthogonalComplement[mat];
Together[Transpose[ocmat].ocmat]
Together[mat.ocmat]

Out[51]= {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}

Out[52]= {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}

Daniel Lichtblau
Wolfram Research

• References:
  • Given a matrix A, construct a semi-orthogonal matrix L
    • From: Tebi <temelt@xs4all.nl>