Vol. 36, No. 3, 1971

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The matrix equation AXB = X

Darald Joe Hartfiel

Vol. 36 (1971), No. 3, 659–669

This paper considers the solutions of the matrix equation AXB = X where we specify A and B to be n-square and doubly stochastic. Solutions are found explicitly and do not depend on either the Jordan or Rational canonical forms. We further find all doubly stochastic solutions of this equation, by noting that Jn = (1∕n), the n-square doubly stochastic matrix in which each entry is 1∕n, is always a solution and that the doubly stochastic solutions form a compact convex set. We solve the equation by characterizing the vertices of this convex set.

Mathematical Subject Classification
Primary: 15.35
Received: 17 February 1970
Published: 1 March 1971
Darald Joe Hartfiel