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Difference between revisions of "Segre characteristic of a square matrix"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301901.png" /> be a square [[Matrix|matrix]] over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301902.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301903.png" />, the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301904.png" />, be an eigenvalue (cf. [[Eigen value|Eigen value]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301905.png" />. Over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301906.png" /> the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301907.png" /> can be put in Jordan normal form (see [[Jordan matrix|Jordan matrix]]). The Segre characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301908.png" /> at the eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s1301909.png" /> is the sequence of sizes of the Jordan blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s13019010.png" /> with eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s13019011.png" /> in non-increasing order. The Segre characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s13019012.png" /> consists of the complete set of data describing the Jordan normal form comprising all eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s13019013.png" /> and the Segre characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s13019014.png" /> at each of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130190/s13019015.png" />.
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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 $\alpha$ and the Segre characteristic of $A$ at each of the $\alpha$.
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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>
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<table>
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<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>
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<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>
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</table>
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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
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article