Linear Transformations on Matrices *
نویسنده
چکیده
Even in thi s generality , it is clear that .!l' (I , ~) is a multiplicative se migroup with an ide ntity. The invariant I can be a scalar valued fun ction, e.g. , I (X) = det (X) ; or for that matter it can describe a property, e.g., m can equal M" (C) and I (X) can mean that X is unitary , so that we are simply asking for the s tructure of all linear transformation s T that map the unitary group into itself. Much of a beginning course in linear algebra is de voted to the study of one as pect of this question for certain choices of I; for example, if I (X) = p (X), the rank of X , then it is well known that the three standard linear operations on the rows and columns of a matrix leave p fixed and this fact permits us to compute p(X) by reducing X to some normal form. The similarity theory is another example of this problem. In this case take I (X) to be the set of all elementary divisors of the characteristic matrix of X, and then the linear operators T that we wish to study are precisely those for which I(X)=/(T(X)). In the survey paper [18, 1962] 1 some of the aspects of this general problem are disc ussed. But since the time that paper was written there have been a number of developments. The purpose of this paper is to describe some of these.
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