### A nested multiplicative commutator equation

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In an invited paper, Baksalary [Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors. In: T. Pukkila, S. Puntanen (Eds.), Proceedings of the Second International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, [2], pp. 113-142] presented 45 necessary and sufficient conditions for the commutativity of a pair of orthogonal projectors. Basing on these results, he discussed therein also statistical aspects of the commutativity...

We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix.

A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we...

Necessary and sufficient conditions are presented for the commutativity equalities ${A}^{*}{A}^{D}={A}^{D}{A}^{*}$, ${A}^{\u2020}{A}^{D}={A}^{D}{A}^{\u2020}$, ${A}^{\u2020}A{A}^{D}={A}^{D}A{A}^{\u2020}$, $A{A}^{D}{A}^{*}={A}^{*}{A}^{D}A$ and so on to hold by using rank equalities of matrices. Some related topics are also examined.

The max algebra consists of the nonnegative real numbers equipped with two binary operations, maximization and multiplication. We characterize the invertible linear operators that preserve the set of commuting pairs of matrices over a subalgebra of max algebra.

Let X and Y be Banach spaces and ℬ(X) and ℬ(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A,B ∈ ℬ(X) quasi-commute if there exists a nonzero scalar ω such that AB = ωBA. We characterize bijective linear maps ϕ : ℬ(X) → ℬ(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

Let $\mathcal{P}$ be an arbitrary parabolic subalgebra of a simple associative $F$-algebra. The ideals of $\mathcal{P}$ are determined completely; Each ideal of $\mathcal{P}$ is shown to be generated by one element; Every non-linear invertible map on $\mathcal{P}$ that preserves ideals is described in an explicit formula.

It is shown that commutativity of two oblique projectors is equivalent with their product idempotency if both projectors are not necessarily Hermitian but orthogonal with respect to the same inner product.