Karl Popper, though not trained as a physicist and embarrassed early in his career by a physics error pointed out by Einstein and Bohr, ultimately made substantial contributions to the interpretation of quantum mechanics. As was often the case, Popper initially formulated his position by criticizing the views of others—in this case Niels Bohr and Werner Heisenberg. Underlying Popper’s criticism...

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Karl Johan Åström & Richard M. Murray: Feedback Systems is published by Princeton University Press and copyrighted, © 2008, by Princeton University Press. All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher, except f...

We characterize matrices A ∈ Cn×n whose zero/nonzero pattern requires that the controllability matrix [b Ab Ab . . . An−1b] ∈ Cn×n is of full rank, where b ∈ Cn×1 has exactly one nonzero entry. When all the diagonal entries of A are nonzero we show that this occurs if and only if QAQ is unreduced upper Hessenberg, with Q being a permutation matrix for which Qb = [b1, 0, . . . , 0] T . We also c...

In this paper, a class of Hessenberg matrices is presented for adoption as test matrices. The Moore-Penrose inverse and the Drazin inverse for each member of this class are determined explicitly.