Difference between revisions of "Minor"
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| − | + | [[Category:Linear and multilinear algebra; matrix theory]] | |
| + | {{MSC|15Axx|}} | ||
| + | {{TEX|done}} | ||
| − | + | ''of order $k$ of a matrix $A$'' | |
| + | This terminology is used (depending upon the context) for | ||
| + | * 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]]''; | ||
| + | * the [[Determinant|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 order $k+1$ 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|cofactor]] expansion of the determinant (see [[Determinant]]) and in the [[Cauchy Binet formula]]. | |
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Latest revision as of 10:05, 20 December 2015
2020 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 order $k+1$ 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.
Minor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minor&oldid=13192