Vol. 33, No. 1, 1970

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 323: 1
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Vol. 317: 1  2
Vol. 316: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Orthogonal groups of positive definite multilinear functionals

Stephen J. Pierce

Vol. 33 (1970), No. 1, 183–189

Let V be a finite dimensional vector space over the real numbers R and let T : V V be a linear transformation. If φ : ×1mV R is a real multilinear functional and

φ(TxI,⋅⋅⋅ ,Txm ) = φ(x1,⋅⋅⋅ ,xm),

x1,,xm V,T is called an isometry with respect to φ. We say φ is positive definite if φ(x,,x) > 0 for all nonzero x V . In this paper we prove that if φ is positive definite and T is an isometry with respect to φ, then all eigenvalues of T have modulus one and all elementary divisors of T over the complex numbers are linear.

Mathematical Subject Classification
Primary: 15.80
Received: 8 July 1960
Published: 1 April 1970
Stephen J. Pierce