Vol. 33, No. 1, 1970

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Orthogonal groups of positive definite multilinear functionals

Stephen J. Pierce

Vol. 33 (1970), No. 1, 183–189
Abstract

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
Milestones
Received: 8 July 1960
Published: 1 April 1970
Authors
Stephen J. Pierce