Editing Minor (linear algebra) (section)

===General definition===

Let {{math|'''A'''}} be an {{math|''m'' × ''n''}} matrix and {{mvar|k}} an [[integer]] with {{math|0 < ''k'' ≤ ''m''}}, and {{math|''k'' ≤ ''n''}}. A {{math|''k'' × ''k''}} ''minor'' of {{math|'''A'''}}, also called ''minor determinant of order {{mvar|k}}'' of {{math|'''A'''}} or, if {{math|''m'' {{=}} ''n''}}, the {{math|(''n'' − ''k'')}}''th'' ''minor determinant'' of {{math|'''A'''}} (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a {{math|''k'' × ''k''}} matrix obtained from {{math|'''A'''}} by deleting {{math|''m'' − ''k''}} rows and {{math|''n'' − ''k''}} columns. Sometimes the term is used to refer to the {{math|''k'' × ''k''}} matrix obtained from {{math|'''A'''}} as above (by deleting {{math|''m'' − ''k''}} rows and {{math|''n'' − ''k''}} columns), but this matrix should be referred to as a ''(square) submatrix'' of {{math|'''A'''}}, leaving the term "minor" to refer to the determinant of this matrix. For a matrix {{math|'''A'''}} as above, there are a total of <math display="inline">{m \choose k} \cdot {n \choose k}</math> minors of size {{math|''k'' × ''k''}}. The ''minor of order zero'' is often defined to be 1. For a square matrix, the ''zeroth minor'' is just the determinant of the matrix.<ref name="Hohn">Elementary Matrix Algebra (Third edition), Franz E. Hohn, The Macmillan Company, 1973, {{isbn|978-0-02-355950-1}}</ref><ref name="Encyclopedia of Mathematics" />

Let 
<math display=block>\begin{align}
  I &= 1 \le i_1 < i_2 < \cdots < i_k \le m, \\[2pt]
  J &= 1 \le j_1 < j_2 < \cdots < j_k \le n,
\end{align}</math>
be ordered sequences (in natural order, as it is always assumed when talking about minors unless otherwise stated) of indexes. The minor <math display="inline">\det \bigl( (\mathbf A_{i_p, j_q})_{p,q = 1, \ldots, k} \bigr)</math> corresponding to these choices of indexes is denoted <math>\det_{I,J} A</math> or <math>\det \mathbf A_{I, J}</math> or <math>[\mathbf A]_{I,J}</math> or <math>M_{I,J}</math> or <math>M_{i_1, i_2, \ldots, i_k, j_1, j_2, \ldots, j_k}</math> or <math>M_{(i),(j)}</math> (where the {{math|(''i'')}} denotes the sequence of indexes {{mvar|I}}, etc.), depending on the source. Also, there are two types of denotations in use in literature: by the minor associated to ordered sequences of indexes {{mvar|I}} and {{mvar|J}}, some authors<ref>Linear Algebra and Geometry, Igor R. Shafarevich,  Alexey O. Remizov, Springer-Verlag Berlin Heidelberg, 2013, {{isbn|978-3-642-30993-9}}</ref> mean the determinant of the matrix that is formed as above, by taking the elements of the original matrix from the rows whose indexes are in {{mvar|I}} and columns whose indexes are in {{mvar|J}}, whereas some other authors mean by a minor associated to {{mvar|I}} and {{mvar|J}} the determinant of the matrix formed from the original matrix by deleting the rows in {{mvar|I}} and columns in {{mvar|J}};<ref name="Hohn" /> which notation is used should always be checked. In this article, we use the inclusive definition of choosing the elements from rows of {{mvar|I}} and columns of {{mvar|J}}. The exceptional case is the case of the first minor or the {{math|(''i'', ''j'')}}-minor described above; in that case, the exclusive meaning <math display="inline">M_{i,j} = \det \bigl( \left( \mathbf A_{p,q} \right)_{p \neq i, q \neq j} \bigr)</math> is standard everywhere in the literature and is used in this article also.