How a Unitoid Matrix Loses Its Unitoidness?

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.