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Difference between revisions of "Minor"

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''of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064100/m0641002.png" />''
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[[Category:Linear and multilinear algebra; matrix theory]]
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{{MSC|15Axx|}}
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{{TEX|done}}
  
A [[Determinant|determinant]] of a [[Matrix|matrix]] whose entries are located in a given matrix at the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064100/m0641003.png" /> distinct columns and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064100/m0641004.png" /> distinct rows. If the row indices and column indices are the same, then the minor is called principal, and if they are the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064100/m0641005.png" /> rows and columns, then it is called a corner. A basic minor of a matrix is any non-zero minor of maximal order. In order that a non-zero minor be basic it is necessary and sufficient that all minors bordering it (that is, minors of an order higher by one and containing it) are equal to zero. The system of rows (columns) of a matrix related to a basic minor form a maximal linearly independent subsystem of the system of all rows (columns) of the matrix.
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''of order $k$ of a matrix $A$''
  
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This terminology is used (depending upon the context) for
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* a $k\times k$ [[Matrix|matrix]] $B$ whose entries are located at the intersection of $k$ distinct columns and $k$ distinct rows of $A$; however a more common terminology for such $B$ is ''square [[Submatrix|submatrix]]'';
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* the [[Determinant|determinant]] of a square submatrix $B$ of $A$.
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The second meaning is the most common and is the one used in the rest of this entry. Instead of  "minor of order k"  one also uses  "minor of degree k".
  
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If the row indices and column indices are the same, then the minor is called principal, and if they are the first $k$ rows and columns, then it is called a corner. A ''basic minor'' of a matrix is the determinant of a square submatrix of maximal order with nonzero determinant. The determinant of a submatrix $C$ of order $k$ is a basic minor if and only if it is nonzero and all submatrices of oder $k+1$ which contain $C$ have zero determinant. The system of rows (columns) of a basic minor form a maximal linearly independent subsystem of the system of all rows (columns) of the matrix.
  
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Minors are used in the [[Cofactor|cofactor]] expansion of the determinant (see [[Determinant]]) and in the [[Cauchy Binet formula]].
Instead of "minor of order k"  one also uses  "minor of degree k" . Sometimes a minor is not understood to be a determinant (as defined above), but the corresponding submatrix (the notion of  "bordering"  uses this interpretation).
 

Revision as of 09:04, 19 August 2013

2010 Mathematics Subject Classification: Primary: 15Axx [MSN][ZBL]

of order $k$ of a matrix $A$

This terminology is used (depending upon the context) for

  • a $k\times k$ matrix $B$ whose entries are located at the intersection of $k$ distinct columns and $k$ distinct rows of $A$; however a more common terminology for such $B$ is square submatrix;
  • the determinant of a square submatrix $B$ of $A$.

The second meaning is the most common and is the one used in the rest of this entry. Instead of "minor of order k" one also uses "minor of degree k".

If the row indices and column indices are the same, then the minor is called principal, and if they are the first $k$ rows and columns, then it is called a corner. A basic minor of a matrix is the determinant of a square submatrix of maximal order with nonzero determinant. The determinant of a submatrix $C$ of order $k$ is a basic minor if and only if it is nonzero and all submatrices of oder $k+1$ which contain $C$ have zero determinant. The system of rows (columns) of a basic minor form a maximal linearly independent subsystem of the system of all rows (columns) of the matrix.

Minors are used in the cofactor expansion of the determinant (see Determinant) and in the Cauchy Binet formula.

How to Cite This Entry:
Minor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minor&oldid=13192
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article