Relative bilinear complexity and matrix multiplication.

The significance of this notion lies, above all, in the key role of matrix multiplication for numerical linear algebra. Thus the following problems all have "exponent' : Matrix inversion, LK-decomposition, evaluation of the determinant or of all coefficients of the characteristic polynomial and for k = C also Qß-decomposition and unitary transformation to Hessenberg form. (See Strassen [48], [49], BunchHopcroft [12], Schönhage [45], Baur-Strassen [3], Keller [32].) Apart from this, such diverse computational problems äs finding the transitive closure of a finite relation, parsing a context free language or Computing a generalized Fourier transform are reducible to matrix multiplication. (See the survey article of Paterson [43], and Beth [4].) Of course the above definition of is incomplete: We have not explained what we mean by an algorithm or by the complexity of a computational problem. Fortunately, there is no need to do so. We may use instead the notion of rank of a bilinear map,

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