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After a brief historical review and an account of the canonical forms attributed to Jordan and Kronecker, a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals. These reductions are by strict equivalence and by congruence, and essentially complete proofs are presented. Some closely related results which can be derived from the canonical forms are also included. They concern simultaneous diagonalization, a new criterion for the existence of positive definite linear combinations of a pair of Hermitian matrices, and the canonical structures of matrices which are self-adjoint in an indefinite inner product.

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