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minor (of a matrix) (Definition)

Given an $ n\times m$ matrix $ A$ with entries $ a_{ij}$, a minor of $ A$ is the determinant of a smaller matrix formed from its entries by selecting only some of the rows and columns. Let $ K=\{k_1,k_2,\ldots,k_p\}$ and $ L=\{l_1,l_2,\ldots,l_p\}$ be subsets of $ \{1,2,\ldots,n\}$ and $ \{1,2,\ldots,m\}$, respectively. The indices are chosen such that $ k_1 < k_2 < \cdots < k_p$ and $ l_1 < l_2 < \cdots < l_p$. The $ p$-th order minor defined by $ K$ and $ L$ is the following determinant

$\displaystyle A\begin{pmatrix}k_1 & k_2 & \cdots & k_p \\ l_1 & l_2 & \cdots & ... ...ots & \vdots \\ a_{k_p l_1} & a_{k_p l_2} & \cdots & a_{k_p k_p} \end{vmatrix}.$    

If $ p$ exceeds either $ m$ or $ n$, then the minor is automatically zero. When $ p=m=n$, the minor is simply the determinant of the matrix. If $ K=L$, then the minor is called principal. The word minor may also refer to just the matrix formed from the selected rows and columns, not necessarily its determinant. The precise meaning is usually clear from context.

There does not seem to be a standard notation for matrix minors. Another possible notation is $ [A]_{K,L}$.

Some authors reserve the term minor for the case when only one row and one column are removed. This use is in conjunction with the concept of a cofactor.



"minor (of a matrix)" is owned by CWoo. [ owner history (1) ]
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See Also: cofactor expansion, Cauchy-Binet formula

Also defines:  principal minor, cofactor
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Cross-references: conjunction, clear, indices, subsets, columns, rows, determinant, matrix
There are 8 references to this entry.

This is version 4 of minor (of a matrix), born on 2004-01-18, modified 2006-04-25.
Object id is 5522, canonical name is MinorOfAMatrix.
Accessed 19954 times total.

Classification:
AMS MSC: 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)
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