Difference between revisions of "Segre characteristic of a square matrix"
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Let $A$ be a square [[matrix]] over a [[field]] $F$ and let $\alpha \in \bar F$, the algebraic closure of $F$, be an [[Eigen value|eigenvalue]] of $A$. Over $\bar F$ the matrix $A$ can be put in [[Jordan normal form]]. The Segre characteristic of $A$ at the eigenvalue $\alpha$ is the sequence of sizes of the Jordan blocks of $A$ with eigenvalue $\alpha$ in non-increasing order. The Segre characteristic of $A$ consists of the complete set of data describing the Jordan normal form comprising all eigenvalues $\alpha_1,\ldots,\alpha_r$ and the Segre characteristic of $A$ at each of the $\alpha_i$. | Let $A$ be a square [[matrix]] over a [[field]] $F$ and let $\alpha \in \bar F$, the algebraic closure of $F$, be an [[Eigen value|eigenvalue]] of $A$. Over $\bar F$ the matrix $A$ can be put in [[Jordan normal form]]. The Segre characteristic of $A$ at the eigenvalue $\alpha$ is the sequence of sizes of the Jordan blocks of $A$ with eigenvalue $\alpha$ in non-increasing order. The Segre characteristic of $A$ consists of the complete set of data describing the Jordan normal form comprising all eigenvalues $\alpha_1,\ldots,\alpha_r$ and the Segre characteristic of $A$ at each of the $\alpha_i$. | ||
Latest revision as of 18:18, 22 November 2016
2020 Mathematics Subject Classification: Primary: 15A21 [MSN][ZBL]
Let $A$ be a square matrix over a field $F$ and let $\alpha \in \bar F$, the algebraic closure of $F$, be an eigenvalue of $A$. Over $\bar F$ the matrix $A$ can be put in Jordan normal form. The Segre characteristic of $A$ at the eigenvalue $\alpha$ is the sequence of sizes of the Jordan blocks of $A$ with eigenvalue $\alpha$ in non-increasing order. The Segre characteristic of $A$ consists of the complete set of data describing the Jordan normal form comprising all eigenvalues $\alpha_1,\ldots,\alpha_r$ and the Segre characteristic of $A$ at each of the $\alpha_i$.
See also: Segre classification.
References
| [a1] | H.W. Turnbull, A.C. Aitken, "An introduction to the theory of canonical matrices" , Blackie (1932) pp. Chapt. VI |
| [a2] | Ch.G. Cullen, "Matrices and linear transformations" , Addison-Wesley (1972) pp. Chap. 5 (Dover reprint, 1990) |
How to Cite This Entry:
Segre characteristic of a square matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_characteristic_of_a_square_matrix&oldid=39802
Segre characteristic of a square matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_characteristic_of_a_square_matrix&oldid=39802
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article