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''of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217201.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217202.png" />''
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''of a matrix  $  A = \| a _ {ij} \| _ {1}  ^ {n} $
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over a field $  K $''
  
 
The polynomial
 
The polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217203.png" /></td> </tr></table>
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$$
 +
p ( \lambda )  = \
 +
\mathop{\rm det} ( A - \lambda E)  = \
 +
\left \|
 +
 
 +
\begin{array}{llll}
 +
a _ {11} - \lambda  &a _ {12}  &\dots  &a _ {1n}  \\
 +
a _ {21}  &a _ {22} - \lambda  &\dots  &a _ {2n}  \\
 +
\dots  &\dots  &\dots  &\dots  \\
 +
a _ {n1}  &a _ {n2}  &\dots  &a _ {nn} - \lambda  \\
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\end{array}
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\
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\right \| =
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217204.png" /></td> </tr></table>
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$$
 +
= \
 +
(- \lambda )  ^ {n} + b _ {1} (- \lambda ) ^ {n - 1 } + \dots + b _ {n}  $$
  
over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217205.png" />. The degree of the characteristic polynomial is equal to the order of the square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217206.png" />, the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217207.png" /> is the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217208.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c0217209.png" />, cf. [[Trace of a square matrix|Trace of a square matrix]]), the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172010.png" /> is the sum of all principal minors of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172011.png" />, in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172012.png" /> (cf. [[Minor|Minor]]). The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172013.png" /> is called the characteristic equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172014.png" /> or the secular equation.
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over $  K $.  
 +
The degree of the characteristic polynomial is equal to the order of the square matrix $  A $,  
 +
the coefficient $  b _ {1} $
 +
is the trace of $  A $(
 +
$  b _ {1} = \mathop{\rm Tr}  A = a _ {11} + \dots + a _ {nn} $,  
 +
cf. [[Trace of a square matrix|Trace of a square matrix]]), the coefficient $  b _ {m} $
 +
is the sum of all principal minors of order $  m $,  
 +
in particular, $  b _ {n} = \mathop{\rm det}  A $(
 +
cf. [[Minor|Minor]]). The equation $  p ( \lambda ) = 0 $
 +
is called the characteristic equation of $  A $
 +
or the secular equation.
  
The roots of the characteristic polynomial lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172015.png" /> are called the characteristic values or eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172016.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172017.png" /> is a number field, then the term  "characteristic number of a matrixcharacteristic numbers"  is also used. Sometimes the roots of the characteristic polynomial are considered in the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172018.png" />. They are usually called the characteristic roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172019.png" />. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172020.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172021.png" /> regarded over an algebraically closed field (for example, over the field of complex numbers) has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172022.png" /> eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172023.png" />, if every root is counted according to its multiplicity. See also [[Eigen value|Eigen value]].
+
The roots of the characteristic polynomial lying in $  K $
 +
are called the characteristic values or eigen values of $  A $.  
 +
If $  K $
 +
is a number field, then the term  "characteristic number of a matrix"  is also used. Sometimes the roots of the characteristic polynomial are considered in the algebraic closure of $  K $.  
 +
They are usually called the characteristic roots of $  A $.  
 +
A matrix $  A $ of order $  n $
 +
regarded over an algebraically closed field (for example, over the field of complex numbers) has $  n $
 +
eigen values $  \lambda _ {1} \dots \lambda _ {n} $,  
 +
if every root is counted according to its multiplicity. See also [[Eigen value|Eigen value]].
  
Similar matrices have the same characteristic polynomial. Every polynomial over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172024.png" /> with leading coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172025.png" /> is the characteristic polynomial of some matrix over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172026.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172027.png" />, the so-called Frobenius matrix.
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Similar matrices have the same characteristic polynomial. Every polynomial over $  K $
 +
with leading coefficient $  (- 1)  ^ {n} $
 +
is the characteristic polynomial of some matrix over $  K $
 +
of order $  n $,  
 +
the so-called Frobenius matrix.
  
 
For references see [[Matrix|Matrix]].
 
For references see [[Matrix|Matrix]].
  
 +
====Comments====
 +
The characteristic roots are often also called the eigen values or characteristic values, thereby not distinguishing the roots of the characteristic polynomial in  $  K $
 +
and those in its algebraic closure. Given a polynomial  $  b ( \lambda ) = ( - \lambda )  ^ {n} + b _ {1} ( - \lambda )  ^ {n-1} + \dots + b _ {n} $.
 +
The matrix in companion form
  
 +
$$
 +
A  =  \left \|
  
====Comments====
+
\begin{array}{lllll}
The characteristic roots are often also called the eigen values or characteristic values, thereby not distinguishing the roots of the characteristic polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172028.png" /> and those in its algebraic closure. Given a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172029.png" />. The matrix in companion form
+
0 & 1  & 0 &\dots  & 0  \\
 
+
0  &\dots  &\dots  &\dots  &\dots  \\
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172030.png" /></td> </tr></table>
+
\dots  &\dots  &\dots  &\dots  &\dots  \\
 +
\dots  &\dots  &\dots  &\dots  & 0  \\
 +
0  &\dots  &\dots  & 0 & 1  \\
 +
b _ {n}  ^  \prime  &\dots  &\dots  &\dots  &b _ {1}  ^  \prime  \\
 +
\end{array}
 +
\right \|
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172031.png" />, has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021720/c02172032.png" /> as its characteristic polynomial.
+
with $  b _ {k}  ^  \prime  = ( - 1 )  ^ {k+1} b _ {k} $,  
 +
has $  b ( \lambda ) $
 +
as its characteristic polynomial.

Latest revision as of 11:43, 24 December 2020


of a matrix $ A = \| a _ {ij} \| _ {1} ^ {n} $ over a field $ K $

The polynomial

$$ p ( \lambda ) = \ \mathop{\rm det} ( A - \lambda E) = \ \left \| \begin{array}{llll} a _ {11} - \lambda &a _ {12} &\dots &a _ {1n} \\ a _ {21} &a _ {22} - \lambda &\dots &a _ {2n} \\ \dots &\dots &\dots &\dots \\ a _ {n1} &a _ {n2} &\dots &a _ {nn} - \lambda \\ \end{array} \ \right \| = $$

$$ = \ (- \lambda ) ^ {n} + b _ {1} (- \lambda ) ^ {n - 1 } + \dots + b _ {n} $$

over $ K $. The degree of the characteristic polynomial is equal to the order of the square matrix $ A $, the coefficient $ b _ {1} $ is the trace of $ A $( $ b _ {1} = \mathop{\rm Tr} A = a _ {11} + \dots + a _ {nn} $, cf. Trace of a square matrix), the coefficient $ b _ {m} $ is the sum of all principal minors of order $ m $, in particular, $ b _ {n} = \mathop{\rm det} A $( cf. Minor). The equation $ p ( \lambda ) = 0 $ is called the characteristic equation of $ A $ or the secular equation.

The roots of the characteristic polynomial lying in $ K $ are called the characteristic values or eigen values of $ A $. If $ K $ is a number field, then the term "characteristic number of a matrix" is also used. Sometimes the roots of the characteristic polynomial are considered in the algebraic closure of $ K $. They are usually called the characteristic roots of $ A $. A matrix $ A $ of order $ n $ regarded over an algebraically closed field (for example, over the field of complex numbers) has $ n $ eigen values $ \lambda _ {1} \dots \lambda _ {n} $, if every root is counted according to its multiplicity. See also Eigen value.

Similar matrices have the same characteristic polynomial. Every polynomial over $ K $ with leading coefficient $ (- 1) ^ {n} $ is the characteristic polynomial of some matrix over $ K $ of order $ n $, the so-called Frobenius matrix.

For references see Matrix.

Comments

The characteristic roots are often also called the eigen values or characteristic values, thereby not distinguishing the roots of the characteristic polynomial in $ K $ and those in its algebraic closure. Given a polynomial $ b ( \lambda ) = ( - \lambda ) ^ {n} + b _ {1} ( - \lambda ) ^ {n-1} + \dots + b _ {n} $. The matrix in companion form

$$ A = \left \| \begin{array}{lllll} 0 & 1 & 0 &\dots & 0 \\ 0 &\dots &\dots &\dots &\dots \\ \dots &\dots &\dots &\dots &\dots \\ \dots &\dots &\dots &\dots & 0 \\ 0 &\dots &\dots & 0 & 1 \\ b _ {n} ^ \prime &\dots &\dots &\dots &b _ {1} ^ \prime \\ \end{array} \right \| $$

with $ b _ {k} ^ \prime = ( - 1 ) ^ {k+1} b _ {k} $, has $ b ( \lambda ) $ as its characteristic polynomial.

How to Cite This Entry:
Characteristic polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characteristic_polynomial&oldid=13190
This article was adapted from an original article by T.S. Pigolkina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article