As is well known, the rank of a diagonalizable complex matrix can be characterized as the maximum order of the nonzero principal minors of this matrix. The standard proof of this fact is based on representing the coefficients of the characteristic polynomial as the (alternating) sums of all the principal minors of appropriate order. We show that in the case of normal matrices, one can give a simple direct proof, not relying on those representations. Bibliography: 2 titles.
Similar content being viewed by others
References
I. V. Proskuryakov, Linear Algebra. Problem Book [in Russian], Nauka, Moscow (1970).
Kh. D. Ikramov, Linear Algebra. Problem Book [in Russian], Lan', St.Petersburg (2006).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 42–44.
Rights and permissions
About this article
Cite this article
Ikramov, K.D. On the ranks of principal submatrices of diagonalizable matrices. J Math Sci 157, 695–696 (2009). https://doi.org/10.1007/s10958-009-9351-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-009-9351-0