# Minor (mathematics)

in linear algebra is a Minor the determinant of a square partial matrix of a matrix.

## (n-1) - reihige Minoren

in connection with the expansion theorem of Laplace orthe matrix inversion means one with Minor of a n × n - matrix A the determinant of a partial matrix, which develops through for capers of a line i and a column j of A. One names this partial matrix with A i, j and the appropriate Minor December (A i, j).

For example the matrix is

[itex] A =
``` \ begin {pmatrix}  1 & 4 & 7 \ \  3 & 0 & 5 \ \ -1 & 9 &11\ end {pmatrix}
```

< to /math> given, and we want to determine the Minor December (A 2.3). Then we cross out the second line in Aand third column:

[itex]
``` \ begin {pmatrix}  1 & 4 & - \ \  - & - & - \ \ -1 & 9 & -\ end {pmatrix}
```

< to /math> and receive

< math> \ December (A_ {2.3}) =
``` \ begin {vmatrix}  1 & 4 \ \ -1 & 9\ end {vmatrix} =9 + 4 = 13.
```

[/itex]

## complementary matrix

with signs provided Minoren

(- 1) i+j·December (A i, j)

are useful with the computation of determinants (see there) and the inverse matrix.

One calls the Transponierte of the matrix consisting of the signed Minoren the matrix A complementary to A#:

[itex] A^ {\ sharp}: =
``` \ begin {pmatrix}         + \ December (A_ {1.1}) & - \ December (A_ {2.1}) & \ ldots & (- 1) ^ {1+n} \ December (A_ {n, 1}) \ \         - \ December (A_ {1.2}) & + \ December (A_ {2.2}) & \ ldots & (- 1) ^ {2+n} \ December (A_ {n, 2}) \ \            \ vdots      &     \ vdots      & \ ddots &         \ vdots          \ \(- 1) ^ {n+1} \ December (A_ {1, n}) & (- 1) ^ {n+2} \ December (A_ {2, n}) & \ ldots & + \ December (A_ {n, n}) \ end {pmatrix}
```

< to /math> Sometimes the complementary matrix becomes also asadjunkte matrix or Adjunkte to A, which symbol names adj (A) or Ad (A). Despite the similarly sounding name it may not be confounded under any circumstances with the adjoints matrix, which marks the complex-conjugated transponierte matrix with symbol A *.

### Characteristics

the complementary matrix too [itex] A< /math> the characteristic has

< math> A \ cdot A^ {\ sharp} = A^ {\ sharp} \ cdot A = \ December (A) \ cdot E< /math>

with [itex] the n \ times n< /math> - unit matrix [itex] E< /math>.

Thus math <\> December (A) is [/itex] invertable, then also A is invertable and it applies

< math> for A^ {- 1} = {1 \ more over \ December (A)}\ A^ cdot {\ sharp}< /math>.

This part could stillfrom the English article w: Adjugate to be supplemented.

## Hauptminoren

the left upper k × k - partial stencils A k of the n × n - matrix A, which results to k lowest lines from cancellation of the n - k rechtesten columns and n -, have one(at least theoretical) meaning for the statement of the definiteness of the matrix A. The determinants December (A k) of these partial stencils are called Hauptminoren.

### Hauptminoren criterion for definiteness

the Hauptminoren criterion for the definiteness (also Hurwitz criterion called) reads as follows:

The symmetrical matrix A is exactly positively definitely if all Hauptminoren of A is positive.

This criterion for stencils up to the format 6×6 is meaningfully applicable.