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Supertropical matrix algebra. (English) Zbl 1215.15018

The objective of this paper is to develop a general algebraic theory of a supertropical matrix algebra, extending previous results of the first author. The main results are as follows:
\(\bullet\)
The tropical determinant (i.e., permanent) is multiplicative when all the determinants involved are tangible.
\(\bullet\)
There exists an adjoint matrix adj\((A)\) such that the matrix \(A\cdot\text{adj}(A)\) behaves much like the identity matrix (times \(|A|\)).
\(\bullet\)
Every matrix \(A\) is a supertropical root of its Hamilton-Cayley polynomial \(f_A\). If these roots are distinct, then \(A\) is conjugate (in a certain supertropical sense) to a diagonal matrix.
\(\bullet\)
The tropical determinant of a matrix \(A\) is a ghost iff the rows of \(A\) are tropically dependent, iff the columns of \(A\) are tropically dependent.
\(\bullet\)
Every root of \(f_A\) is a “supertropical” eigenvalue of \(A\) (appropriately defined), and has a tangible supertropical eigenvector.

MSC:

15A30 Algebraic systems of matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors

Software:

PlanetMath
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References:

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