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A matrix identity and its applications. (English) Zbl 1310.15023

Summary: We first prove a matrix identity concerning the blocks of generalized Jordan blocks and then give applications to some invariants of matrices. As a consequence, we reprove the well known fact that for an eigenvalue \(\lambda\), its algebraic multiplicity is greater than or equal to its geometric multiplicity.

MSC:

15A24 Matrix equations and identities
15A21 Canonical forms, reductions, classification
15A18 Eigenvalues, singular values, and eigenvectors
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References:

[1] Bhattacharya, P. B.; Jain, S. K.; Nagpaul, S. R., Basic Abstract Algebra (1994), Cambridge University Press · Zbl 0837.00002
[2] Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press · Zbl 0704.15002
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