Difference between revisions of "Segre characteristic of a square matrix"
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| − | Let | + | 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 $\alpha$ and the Segre characteristic of $A$ at each of the $\alpha$. |
| + | |||
| + | See also: [[Segre classification]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H.W. Turnbull, A.C. Aitken, "An introduction to the theory of canonical matrices" , Blackie (1932) pp. Chapt. VI</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Ch.G. Cullen, "Matrices and linear transformations" , Addison-Wesley (1972) pp. Chap. 5 (Dover reprint, 1990)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H.W. Turnbull, A.C. Aitken, "An introduction to the theory of canonical matrices" , Blackie (1932) pp. Chapt. VI</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> Ch.G. Cullen, "Matrices and linear transformations" , Addison-Wesley (1972) pp. Chap. 5 (Dover reprint, 1990)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 18:15, 22 November 2016
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 $\alpha$ and the Segre characteristic of $A$ at each of the $\alpha$.
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=39801
Segre characteristic of a square matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_characteristic_of_a_square_matrix&oldid=39801
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article