On Stabilizing Matrices by Simple Row Operations

by Jon H. Folkman

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An affirmative answer is given to the following question: Let A be a nonsingular matrix. Are there matrices P and Q, each a product of diagonal and permutation matrices, such tha PAQ is stable; i.e., all of its eigenvalues have negative real part? This question arises in attempting to solve the equation Ax = b with an analog computer. The problem of finding a computationally practical method of doing what this Memorandum shows can be done remains open. An affirmative answer is given to the following question: Let A be a nonsingular matrix. Are there matrices P and Q, each a product of diagonal and permutation matrices, such that PAQ is stable; i.e., all of its eigenvalues have negative real part? This question arises in attempting to solve the equation Ax = b with an analog computer. The result is an existence theorem rather than an effective algorithm. The problem of finding a computationally practical method of doing what this Memorandum shows can be done remains open.

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