abs_vol20_pp436-448. 25KB Jun 04 2011 12:05:48 AM

ELA

THE LINEAR INDEPENDENCE OF SETS OF TWO AND THREE

CANONICAL ALGEBRAIC CURVATURE TENSORS

∗

ALEXANDER DIAZ† _{AND COREY DUNN}‡

Abstract. We generalize the construction of canonical algebraic curvature tensors by self-adjoint endomorphisms of a vector space to arbitrary endomorphisms. Provided certain basic rank requirements are met, we establish a converse of the classical fact that ifAis symmetric, thenR_A is an algebraic curvature tensor. This allows us to establish a simultaneous diagonalization result in the event that three algebraic curvature tensors are linearly dependent. We use these results to establish necessary and sufficient conditions that a set of two or three algebraic curvature tensors be linearly independent. We present the proofs of these results using elementary methods.

Key words. Algebraic curvature tensor, Linear independence, Simultaneous diagonalization.

AMS subject classifications.15A21, 15A63.

∗_{Received by the editors January 28, 2010. Accepted for publication July 30, 2010. Handling}

Editor: Raphael Loewy.

†_{Mathematics Department, Sam Houston State University, Huntsville, TX 77341, USA}

(acd008@shsu.edu).

‡_{Mathematics Department, California State University at San Bernardino, San Bernardino, CA}

92407, USA (cmdunn@csusb.edu). (Corresponding author). Electronic Journal of Linear Algebra ISSN 1081-3810

A publication of the International Linear Algebra Society Volume 20, pp. 436-448, August 2010