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D.A. Harville, Matrix Algebra From a Statistician’s Perspective, Springer Science+Business Media, LLC, New York, 2008. Full Rank Factorizations. Theorem 1 (Harville, Theorem 4.4.8). Let A represent an m×n nonnull matrix of rank r. Then, there exist an m × r matrix B and an r × n matrix T such that A = BT . Moreover, for any m × r matrix B and r × n matrix T such that A = BT , rank(B) = rank(T )= r, that is, B has full column rank and T has full row rank. Example 2. Consider the matrix A = 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ∼ 1 0 -2 0 0 1 1 0 0 0 0 1 0 0 0 0 = B. B is in reduced echelon form. Then C is obtained by removing the third column of A, the only one which is not a pivot column, and F by getting rid of the last row of zeroes, so C = 1 3 4 2 7 9 1 5 1 1 2 8 , F = 1 0 -2 0 0 1 1 0 0 0 0 1 . It is straightforward to check that A = 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 = 1 3 4 2 7 9 1 5 1 1 2 8 1 0 -2 0 0 1 1 0 0 0 0 1 = CF. Explicit Formula for A † . C.C. MacDuffee apparently was the first to point out, in private commu- nications about 1959, that a full-rank factorization of a matrix A leads to an explicit formula for its Moore-Penrose inverse, A † . We now show the existence. Let A = FG be a rank factorization. Then it can be easily verified that F † =(F * F ) -1 F * , G † = G * (GG * ) -1 and then A † = G † F † . Theorem 3 (MacDuffee). If A ∈ C m×n r , r> 0, has a full-rank factorization A = F G, (1) then A † = G * (F * AG * ) -1 F * . [= G * (GG * ) -1 (F * F ) -1 F * .] (2)
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D.A. Harville, Matrix Algebra From a Statistician’s Perspective, SpringerScience+Business Media, LLC, New York, 2008.
Full Rank Factorizations.
Theorem 1 (Harville, Theorem 4.4.8). Let A represent an m×n nonnull matrixof rank r. Then, there exist an m × r matrix B and an r × n matrix T suchthat A = BT . Moreover, for any m× r matrix B and r× n matrix T such thatA = BT , rank(B) = rank(T ) = r, that is, B has full column rank and T hasfull row rank.
Example 2. Consider the matrix
A =
1 3 1 42 7 3 91 5 3 11 2 0 8
∼
1 0 −2 00 1 1 00 0 0 10 0 0 0
= B.
B is in reduced echelon form. Then C is obtained by removing the third columnof A, the only one which is not a pivot column, and F by getting rid of the lastrow of zeroes, so
C =
1 3 42 7 91 5 11 2 8
, F =
1 0 −2 00 1 1 00 0 0 1
.
It is straightforward to check that
A =
1 3 1 42 7 3 91 5 3 11 2 0 8
=
1 3 42 7 91 5 11 2 8
1 0 −2 00 1 1 00 0 0 1
= CF.
Explicit Formula for A†.C.C. MacDuffee apparently was the first to point out, in private commu-
nications about 1959, that a full-rank factorization of a matrix A leads to anexplicit formula for its Moore-Penrose inverse, A†.
We now show the existence. Let A = FG be a rank factorization. Then itcan be easily verified that
F † = (F ∗F )−1F ∗, G† = G∗(GG∗)−1
and thenA† = G†F †.
Theorem 3 (MacDuffee). If A ∈ Cm×nr , r > 0, has a full-rank factorization
A = FG, (1)
thenA† = G∗(F ∗AG∗)−1F ∗. [= G∗(GG∗)−1(F ∗F )−1F ∗.] (2)
