Abstract
A unitoid is a square matrix that can be brought to diagonal form by a congruence transformation. Among different diagonal forms of a unitoid \(A\) , there is only one, up to the order adopted for the principal diagonal, whose nonzero diagonal entries all have the modulus 1. It is called the congruence canonical form of \(A\) , while the arguments of the nonzero diagonal entries are called the canonical angles of \(A\) . If \(A\) is nonsingular then its canonical angles are closely related to the arguments of the eigenvalues of the matrix \(A^{-*}A\) , called the cosquare of \(A\) . Although the definition of a unitoid reminds the notion of a diagonalizable matrix in the similarity theory, the analogy between these two matrix classes is misleading. We show that the Jordan block \(J_{n}(1)\) , which is regarded as an antipode of diagonalizability in the similarity theory, is a unitoid. Moreover, its cosquare \(C_{n}(1)\) has \(n\) distinct unimodular eigenvalues. Then we immerse \(J_{n}(1)\) in the family of the Jordan blocks \(J_{n}(\lambda)\) , where \(\lambda\) is varying in the range \((0,2]\) . At some point to the left of 1, \(J_{n}(\lambda)\) is not a unitoid any longer. We discuss this moment in detail in order to comprehend how it can happen. Similar moments with even smaller \(\lambda\) are discussed, and certain remarkable facts about the eigenvalues of cosquares and their condition numbers are pointed out.
