How a unitoid matrix loses its unitoidness?
Kh. D. Ikramova,
A. M. Nazarib a Lomonosov Moscow State University, Russia
b Arak University, Iran
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.
Key words:
congruence transformation, canonical form, cosquare, unitoid matrix, canonical angles.
UDC:
512.643 Received: 01.11.2023
Revised: 27.11.2023
Accepted: 19.04.2024
DOI:
10.15372/SJNM20240302
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