# THE NATURAL RIGHT AND THE NATURAL LEFT INVERSES OF RECTANGULAR MATRICES

## Authors

• ALEXANDER ABIAN Department of Mathematics, Iowa State Uruversity, Ames, Iowa 50011, U.S.A.
• PAULA A. KEMP Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65803, U.S.A.

## Keywords:

THE NATURAL RIGHT AND THE NATURAL LEFT INVERSES OF RECTANGULAR MATRICES

## Abstract

If an $m$ by $n$ with $m < n$ matrix $A$ has a right inverse then it has infinitely many right inverses. In fact, $K (AK )^{-1}$ is a right inverse of $A$ for many $n$ by $m$ matrices $K$ of rank $m$. The natural choice for $K$ is the transpose $A'$ of $A$. Thus, we call $A'(AA')^{-1}$ the natural right inverse of $A$. It can be used (not so obviously) to solve $AX = C$ yielding the solution $X = A'(AA')^{-1}C$ which minimizes the length $||X||$. Similarly, if an $n$ by $m$ with $m < n$ matrix $B$ has a left inverse, we call $(B'B)^{-1}B'$ the natural left inverse of $B$. It can be used (in an obvious way) in an attempt "to solve" $BX =C$ yielding the best approximate solution $X =(B'B)^{-1}B'C$ which minimizes the error $|| BX||$.

## References

Bronson, R. "Matrix Methods". Academic Press, New York, 1970.

Abian, A. "The Natural Right and the Natural Left Inverses of Rectangular Matrices". Amer. Math. Soc. Abstracts 5(1984) 394.

1990-12-01

## How to Cite

ABIAN, A., & KEMP, P. A. (1990). THE NATURAL RIGHT AND THE NATURAL LEFT INVERSES OF RECTANGULAR MATRICES. Tamkang Journal of Mathematics, 21(3), 279–286. https://doi.org/10.5556/j.tkjm.21.1990.4690

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