Minor (linear algebra)

In linear algebra, a minor of a matrix Template:Math is the determinant of some smaller square matrix generated from Template:Math by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition.

Definition and illustration

First minors

If Template:Math is a square matrix, then the minor of the entry in the Template:Mvar-th row and Template:Mvar-th column (also called the Template:Math minor, or a first minor<ref>Burnside, William Snow & Panton, Arthur William (1886) Theory of Equations: with an Introduction to the Theory of Binary Algebraic Form.</ref>) is the determinant of the submatrix formed by deleting the Template:Mvar-th row and Template:Mvar-th column. This number is often denoted Template:Math. The Template:Math cofactor is obtained by multiplying the minor by Template:Math.

To illustrate these definitions, consider the following Template:Nowrap matrix,

<math display=block>\begin{bmatrix} 1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ \end{bmatrix}</math>

To compute the minor Template:Math and the cofactor Template:Math, we find the determinant of the above matrix with row 2 and column 3 removed.

<math display=block> M_{2,3} = \det \begin{bmatrix}

 1    & 4    & \Box \\
 \Box & \Box & \Box \\
 -1   & 9    & \Box \\

\end{bmatrix}= \det \begin{bmatrix}

 1 & 4 \\
 -1 & 9 \\

\end{bmatrix} = 9-(-4) = 13</math>

So the cofactor of the Template:Nowrap entry is

<math display=block>C_{2,3} = (-1)^{2+3}(M_{2,3}) = -13.</math>

General definition

Let Template:Math be an Template:Math matrix and Template:Mvar an integer with Template:Math, and Template:Math. A Template:Math minor of Template:Math, also called minor determinant of order Template:Mvar of Template:Math or, if Template:Math, the Template:Mathth minor determinant of Template:Math (the word "determinant" is often omitted, and the word "degree" is sometimes used instead of "order") is the determinant of a Template:Math matrix obtained from Template:Math by deleting Template:Math rows and Template:Math columns. Sometimes the term is used to refer to the Template:Math matrix obtained from Template:Math as above (by deleting Template:Math rows and Template:Math columns), but this matrix should be referred to as a (square) submatrix of Template:Math, leaving the term "minor" to refer to the determinant of this matrix. For a matrix Template:Math as above, there are a total of <math display="inline">{m \choose k} \cdot {n \choose k}</math> minors of size Template:Math. 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, Template:Isbn</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 Template:Math denotes the sequence of indexes Template:Mvar, 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 Template:Mvar and Template:Mvar, some authors<ref>Linear Algebra and Geometry, Igor R. Shafarevich, Alexey O. Remizov, Springer-Verlag Berlin Heidelberg, 2013, Template:Isbn</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 Template:Mvar and columns whose indexes are in Template:Mvar, whereas some other authors mean by a minor associated to Template:Mvar and Template:Mvar the determinant of the matrix formed from the original matrix by deleting the rows in Template:Mvar and columns in Template:Mvar;<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 Template:Mvar and columns of Template:Mvar. The exceptional case is the case of the first minor or the Template:Math-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.

Complement

The complement Template:Math of a minor Template:Math of a square matrix, Template:Math, is formed by the determinant of the matrix Template:Math from which all the rows (Template:Mvar) and columns (Template:Mvar) associated with Template:Math have been removed. The complement of the first minor of an element Template:Mvar is merely that element.<ref>Bertha Jeffreys, Methods of Mathematical Physics, p.135, Cambridge University Press, 1999 Template:Isbn.</ref>

Applications of minors and cofactors

Cofactor expansion of the determinant

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The cofactors feature prominently in Laplace's formula for the expansion of determinants, which is a method of computing larger determinants in terms of smaller ones. Given an Template:Math matrix Template:Math, the determinant of Template:Math, denoted Template:Math, can be written as the sum of the cofactors of any row or column of the matrix multiplied by the entries that generated them. In other words, defining <math>C_{ij} = (-1)^{i+j} M_{ij}</math> then the cofactor expansion along the Template:Mvar-th column gives:

<math display=block>\begin{align}

 \det(\mathbf A) &= a_{1j}C_{1j} + a_{2j}C_{2j} + a_{3j}C_{3j} + \cdots + a_{nj}C_{nj} \\[2pt]
 &= \sum_{i=1}^{n} a_{ij} C_{ij} \\[2pt]
 &= \sum_{i=1}^{n} a_{ij}(-1)^{i+j} M_{ij}

\end{align}</math>

The cofactor expansion along the Template:Mvar-th row gives:

<math display=block>\begin{align}

 \det(\mathbf A) &= a_{i1}C_{i1} + a_{i2}C_{i2} + a_{i3}C_{i3} + \cdots + a_{in}C_{in} \\[2pt]
 &= \sum_{j=1}^{n} a_{ij} C_{ij} \\[2pt]
 &= \sum_{j=1}^{n} a_{ij} (-1)^{i+j} M_{ij}

\end{align}</math>

Inverse of a matrix

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

One can write down the inverse of an invertible matrix by computing its cofactors by using Cramer's rule, as follows. The matrix formed by all of the cofactors of a square matrix Template:Math is called the cofactor matrix (also called the matrix of cofactors or, sometimes, comatrix):

<math display=block>\mathbf C = \begin{bmatrix}

 C_{11}  & C_{12} & \cdots &   C_{1n} \\
 C_{21}  & C_{22} & \cdots &   C_{2n} \\
  \vdots & \vdots & \ddots & \vdots \\
 C_{n1}  & C_{n2} & \cdots &  C_{nn}

\end{bmatrix} </math>

Then the inverse of Template:Math is the transpose of the cofactor matrix times the reciprocal of the determinant of Template:Math:

<math display=block>\mathbf A^{-1} = \frac{1}{\operatorname{det}(\mathbf A)} \mathbf C^\mathsf{T}.</math>

The transpose of the cofactor matrix is called the adjugate matrix (also called the classical adjoint) of Template:Math.

The above formula can be generalized as follows: Let <math display=block>\begin{align}

 I &= 1 \le i_1 < i_2 < \ldots < i_k \le n, \\[2pt]
 J &= 1 \le j_1 < j_2 < \ldots < j_k \le n,

\end{align}</math> be ordered sequences (in natural order) of indexes (here Template:Math is an Template:Math matrix). Then<ref name="Prasolov1994">Template:Cite book</ref>

<math display=block>[\mathbf A^{-1}]_{I,J} = \pm\frac{[\mathbf A]_{J',I'}}{\det \mathbf A},</math>

where Template:Math denote the ordered sequences of indices (the indices are in natural order of magnitude, as above) complementary to Template:Math, so that every index Template:Math appears exactly once in either Template:Mvar or Template:Mvar, but not in both (similarly for the Template:Mvar and Template:Mvar) and Template:Math denotes the determinant of the submatrix of Template:Math formed by choosing the rows of the index set Template:Mvar and columns of index set Template:Mvar. Also, <math>[\mathbf A]_{I,J} = \det \bigl( (A_{i_p, j_q})_{p,q = 1, \ldots, k} \bigr).</math> A simple proof can be given using wedge product. Indeed,

<math display=block>\bigl[ \mathbf A^{-1} \bigr]_{I,J} (e_1\wedge\ldots \wedge e_n) = \pm(\mathbf A^{-1}e_{j_1})\wedge \ldots \wedge(\mathbf A^{-1}e_{j_k})\wedge e_{i'_1}\wedge\ldots \wedge e_{i'_{n-k}}, </math>

where <math>e_1, \ldots, e_n</math> are the basis vectors. Acting by Template:Math on both sides, one gets

<math display=block>\begin{align}

 &\ \bigl[\mathbf A^{-1} \bigr]_{I,J} \det \mathbf A (e_1\wedge\ldots \wedge e_n) \\[2pt]
 =&\ \pm (e_{j_1})\wedge \ldots \wedge(e_{j_k})\wedge (\mathbf A e_{i'_1})\wedge\ldots \wedge (\mathbf A e_{i'_{n-k}}) \\[2pt]
 =&\ \pm [\mathbf A]_{J',I'}(e_1\wedge\ldots \wedge e_n).

\end{align}</math>

The sign can be worked out to be <math display=block>(-1)^\wedge \!\!\left( \sum_{s=1}^{k} i_s - \sum_{s=1}^{k} j_s \right),</math> so the sign is determined by the sums of elements in Template:Mvar and Template:Mvar.

Other applications

Given an Template:Math matrix with real entries (or entries from any other field) and rank Template:Mvar, then there exists at least one non-zero Template:Math minor, while all larger minors are zero.

We will use the following notation for minors: if Template:Math is an Template:Math matrix, Template:Mvar is a subset of Template:Math with Template:Mvar elements, and Template:Mvar is a subset of Template:Math with Template:Mvar elements, then we write Template:Math for the Template:Math minor of Template:Math that corresponds to the rows with index in Template:Mvar and the columns with index in Template:Mvar.

• If Template:Math, then Template:Math is called a principal minor.
• If the matrix that corresponds to a principal minor is a square upper-left submatrix of the larger matrix (i.e., it consists of matrix elements in rows and columns from 1 to Template:Mvar, also known as a leading principal submatrix), then the principal minor is called a leading principal minor (of order Template:Mvar) or corner (principal) minor (of order Template:Mvar).<ref name="Encyclopedia of Mathematics">Template:Cite book</ref> For an Template:Math square matrix, there are Template:Mvar leading principal minors.
• A basic minor of a matrix is the determinant of a square submatrix that is of maximal size with nonzero determinant.<ref name="Encyclopedia of Mathematics" />
• For Hermitian matrices, the leading principal minors can be used to test for positive definiteness and the principal minors can be used to test for positive semidefiniteness. See Sylvester's criterion for more details.

Both the formula for ordinary matrix multiplication and the Cauchy–Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that Template:Math is an Template:Math matrix, Template:Math is an Template:Math matrix, Template:Mvar is a subset of Template:Math with Template:Mvar elements and Template:Mvar is a subset of Template:Math with Template:Mvar elements. Then <math display=block>[\mathbf{AB}]_{I,J} = \sum_{K} [\mathbf{A}]_{I,K} [\mathbf{B}]_{K,J}\,</math> where the sum extends over all subsets Template:Mvar of Template:Math with Template:Mvar elements. This formula is a straightforward extension of the Cauchy–Binet formula.

Multilinear algebra approach

A more systematic, algebraic treatment of minors is given in multilinear algebra, using the wedge product: the Template:Mvar-minors of a matrix are the entries in the Template:Mvar-th exterior power map.

If the columns of a matrix are wedged together Template:Mvar at a time, the Template:Math minors appear as the components of the resulting Template:Mvar-vectors. For example, the 2 × 2 minors of the matrix <math display=block>\begin{pmatrix}

 1 & 4 \\
 3 & \!\!-1 \\
 2 & 1 \\

\end{pmatrix}</math> are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product <math display=block>(\mathbf{e}_1 + 3\mathbf{e}_2 + 2\mathbf{e}_3) \wedge (4\mathbf{e}_1 - \mathbf{e}_2 + \mathbf{e}_3)</math> where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is bilinear and alternating, <math display=block>\mathbf{e}_i \wedge \mathbf{e}_i = 0,</math> and antisymmetric, <math display=block>\mathbf{e}_i\wedge \mathbf{e}_j = - \mathbf{e}_j\wedge \mathbf{e}_i,</math> we can simplify this expression to <math display=block> -13 \mathbf{e}_1\wedge \mathbf{e}_2 -7 \mathbf{e}_1\wedge \mathbf{e}_3 +5 \mathbf{e}_2\wedge \mathbf{e}_3</math> where the coefficients agree with the minors computed earlier.

A remark about different notation

In some books, instead of cofactor the term adjunct is used.<ref>Felix Gantmacher, Theory of matrices (1st ed., original language is Russian), Moscow: State Publishing House of technical and theoretical literature, 1953, p.491,</ref> Moreover, it is denoted as Template:Math and defined in the same way as cofactor: <math display=block>\mathbf{A}_{ij} = (-1)^{i+j} \mathbf{M}_{ij}</math>

Using this notation the inverse matrix is written this way: <math display=block>\mathbf{M}^{-1} = \frac{1}{\det(M)}\begin{bmatrix}

 A_{11}  & A_{21} & \cdots &   A_{n1}   \\
 A_{12}  & A_{22} & \cdots &   A_{n2}   \\
  \vdots & \vdots & \ddots & \vdots \\
 A_{1n}  & A_{2n} & \cdots &  A_{nn}

\end{bmatrix} </math>

Keep in mind that adjunct is not adjugate or adjoint. In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator.

See also

• Submatrix
• Compound matrix

External links

• MIT Linear Algebra Lecture on Cofactors at Google Video, from MIT OpenCourseWare
• PlanetMath entry of Cofactors
• Springer Encyclopedia of Mathematics entry for Minor

{{#invoke:Navbox|navbox}}