Matrix (mathematics)

in linear algebra is a matrix (Plural: Stencils) an arrangement of numbers (or other objects) in tabular form. One speaks of the columns and lines of the matrix; they form line and/or. Columns vectors. The elements, which are arranged in the matrix,one calls entries or components of the matrix.

The designation „matrix “ was introduced 1850 by James Joseph Sylvester. Stencils represent connections, in which in particular linear combinations play a role, clearly and facilitate thereby computing and thought procedures.

Table of contents 1 notation and first characteristics 2 addition and multiplication 2,1 stencil addition 2,2 scalar multiplication 2,3 stencil multiplication 3 further arithmetic operations 3,1 strengthening stencils 3,2 inverse matrix of 3,3 vector cross products (dot product and tensor product) 3,4 the transponierte matrix 4 uses 4,1 connection with linear illustrations 4,2 transforming stencil equations 5 special stencils 5,1 characteristics of Endomorphismen 5,2 characteristicsfrom Bilinearformen notation and first characteristics than

the lining up of the elements

in lines and columns became generally accepted 5,3 further constructions 6 Verallgemeinerungen [work on] with a large opening and right parenthesis; e.g. the notation stands

< math>

\ begin {pmatrix} a_ {11} & a_ {12} & a_ {13}\ \ a_ {21} & a_ {22} & a_ {23} \ end {pmatrix}

< to /math> for a matrix with 2 lines and 3 columns. More exactly specified one speaks of one <math> m \ times n </math> - matrix with <math> m </math> Lines and <math> n </math> Columns. Therefore one calls also <math> m</math> the line dimension and <math> n </math> the column dimension of the matrix.

A matrix a function is formal

<math> A:\ {1, \ ldots, m \} \ times \ {1, \ dots, n \} \ tons of K, \ quad (i, j) \ mapsto A (i, j) =a_ {ij}< /math>,

pairs of indices <math> (i, j)< /math> Entries <math> a_ {ij}< /math> assigns. The natural numbers are <math> m, n \ geq 1< /math> like above line and column number and <math> K< /math> any quantity. Notto confound with this formal definition of a matrix by functions it is that some stencils themselves define linear illustrations.

Mostly we have to do it with the case that <math> K= \ Bbb Z, \ Bbb Q, \ Bbb R, \ Bbb C< /math> a number set is. In the mathematical subsection of algebra often stencils become alsoEntries in a ring regards.

The quantity <math> \ operator name {Abb} (\ {1, \ ldots, m \} \ times \ {1, \ ldots, n \}, K)< /math> everything <math> m \ times n </math> - stencils over the quantity <math> K< /math> becomes in usual mathematical notation also <math> K^ {\ {1, \ ldots, m \} \ times \ {1, \ ldots n \}}< /math> written; for this the short notation has itself <math> K^ {m \ times n}< /math> in-patriated (sometimes also the way of writing becomes <math> K^ {m, n} </math>, more rarely <math>\, ^m K^n </math> used).

Is <math> A< /math> one <math> m \ times n </math> - matrix over <math> K< /math>, then become the individual function values of pairs of indices (components) in place of the function way of writing <math> A (i, j)< /math> with the index notation <math> a_ {i, j}< /math> or still more briefly <math> a_ {ij}< /math> written. Tabulates one the components in line/, Then one receives the way of writing to column arrangement

< math> A=

 \ begin {pmatrix}  a_ {11} & a_ {12} & \ cdots & a_ {1n} \ \  a_ {21} & a_ {22} & \ cdots & a_ {2n} \ \  \ vdots & \ vdots & \ ddots & \ vdots \ \  a_ {m1} & a_ {m2} & \ dots & a_ {mn}\ end {pmatrix}

< to /math>

The component<math> a_ {ij}< /math> stands thus in <math> i </math> - ten line and <math> j </math> - ten column.

If line and column dimension agree, then one speaks of a square matrix.

If a matrix has only one column, then one calls it a column vector; it has only oneLine, then one calls her a line vector. (That is a shortened, inaccurate manner of speaking, because a einspaltige or single-line matrix can be only one representation of a vector, dependent on the coordinate system - contrary to the vector.) a vector out <math> K^n </math> one can everafter context as single-line or einspaltige matrix represent (thus as element out <math> K^ {1 \ times n} </math> or <math> K^ {n \ times 1} </math>). The use of a column vector has the advantage that one can multiply these directly by a suitable matrix.

additionand multiplication

stencil addition

the sum of two <math> m \ times n< /math> - stencils computes itself, by adding the entries of the two stencils in each case:

<math> A+B = (a_ {ij} +b_ {ij}) _ {i=1, \ ldots, m; \ j=1, \ ldots, n}< /math>

Arithmetical example:

<math>

\ begin {pmatrix} 1 & 3 & 2 \ \1 & 2 & 2 \ end {pmatrix} + \ begin {pmatrix} to 0 & 0 & 5 \ \ 2 & 1 & 1 \ end {pmatrix} = \ begin {pmatrix} to 1+0 & 3+0 & 2+5 \ \ 1+2 & 2+1 & 2+1 \ end {pmatrix} = \ begin {pmatrix} to 1 & 3 & 7 \ \ 3 & 3& 3 \ end {pmatrix}

< to /math>

Only stencils with the same number of lines and columns can be added.

In linear algebra the entries of the stencils are usually elements of a body, like e.g. the real or complex numbers. In this case is thoseStencil addition associatively, commutatively and possesses with the zero-matrix (a matrix of them all entries <math> 0< /math> is) a neutral element. Generally the stencil addition possesses these characteristics however only, if the entries are elements of an algebraic structure, which has these characteristics.

scalar multiplication

oneMatrix is multiplied by a scalar, as all entries of the matrix are multiplied by the scalar:

<math> \ lambda \ A = (\ lambda \ cdot a_ {ij}) cdot _ {i=1, \ ldots, m; \ j=1, \ ldots, n}< /math>

Arithmetical example:

<math> to 2 \ cdot

 \ begin {pmatrix} 1 & 3 & 2 \ \ 1 & 2 & 2 \ end {pmatrix}= \ begin {pmatrix} 2 \ cdot1 & 2 \ cdot3 & 2 \ cdot2 \ \ 2 \ cdot1 & 2 \ cdot2 & 2 \ cdot2 \ end {pmatrix} = \ begin {pmatrix} to 2 & 6 & 4 \ \ 2 & 4 & 4 \ end {pmatrix}

< to /math>

The scalar multiplication may not be confounded with the dot product. Overthe scalar multiplication to accomplish the scalar math <\> lambda /math< must > and the entries of the matrix the same ring <math> (K, +, \ cdot, 0)< /math> come of. The quantity <math> of the m \ times n< /math> - stencils is in this case <math> a R< /math> - (left) module over <math> R< /math>.

stencil multiplication

two stencils are multiplied, by the product sum formula similarly the dot product upPairs from a line vector first and a column vector of the second matrix one uses:

<math> OFF = (c_ {ij}) _ {i=1 \ ldots l, \; j=1 \ ldots n}< /math> and <math> c_ {ij} = \ sum_ {k=1} ^m a_ {IC} \ cdot b_ {kj}< /math>

Arithmetical example:

<math>

\ begin {pmatrix} 1 & 2 & 3 \ \ 4 & 5 & 6 \ \ \ end {pmatrix} \ cdot \ begin {pmatrix} to 6 & -1\ \ 3 & 2 \ \ 0 & -3 \ end {pmatrix} = \ begin {pmatrix} to 1 \ cdot 6 + 2 \ cdot 3 + 3 \ cdot 0 & 1 \ cdot (- 1) + 2 \ cdot 2 + 3 \ cdot (- 3) \ \ 4 \ cdot 6 + 5 \ cdot 3 + 6 \ cdot0 & 4 \ cdot (- 1) + 5 \ cdot 2 + 6 \ cdot (- 3) \ \ \ end {pmatrix} = \ begin {pmatrix} 12 & -6 \ \ 39 & -12 \ end {pmatrix}

< to /math>

With the computation the Falk pattern offers an assistance by hand. It pay attention that stencil multiplication generally not commutative is, i.e. generally math <\> mathrm {B applies \ cdot for A} \ neq \ mathrm {A \ cdot B}< /math>. On the other hand the stencil multiplication is associative in each case: <math> \ mathrm {(A \ cdot B) \ C cdot}< /math> = <math> \ mathrm {A \ cdot (B \ cdot C)}</math>

Around two stencils to multiply the entries a ring must come of and the number of columns to the left with the number of lines of the right matrix agree. Is now <math> A< /math> one <math> l \ times m< /math> - matrix and <math> B< /math> one <math> m \ times n< /math> - matrix then is <math> A \ cdot B< /math> one <math> l \ times n< /math> - matrix.

The square stencils play a special role concerning the stencil multiplicationover a ring <math> R< /math>. These form even with the stencil addition and - multiplication again a ring. The ring is <math> R< /math> unitarily with the one element 1, then the unit matrix is

<math> E =

 \ begin {pmatrix} 1 & 0 & \ cdots & 0 \ \ 0 & 1 & \ cdots &0 \ \ \ vdots & \ vdots & \ ddots & \ vdots \ \ 0 & 0 & \ dots & 1 \ end {pmatrix}

< to /math> the one element of the stencil ring, i.e. this is also unitary. However is the stencil ring <math> K^ {n \ times n} </math> for <math> n> 1< /math> never commutatively.

Further arithmetic operations

strengthening stencils

square stencils <math> A \ in K^ {n \ times n}< /math> can be multiplied by itself; similarly to the case of the real numbers one leads the shortening power mode <math> A^2=A \ cdot A< /math> or <math> A^3=A \ cdot A \ cdot A< /math> etc. . Thus it is also meaningful, square stencilsto begin as elements in polynomials. For large remarks for this see characteristic polynomial.

inverse matrix

major item: Inverse matrix

for some square stencils <math> A< /math> there is an inverse matrix <math> A ^ {- 1}< /math> to math

< A> applies \ cdot for A^ {- 1} = A^ {- 1} \ cdot A = E< /math>

whereby <math> E< /math> the unit matrix is. One calls stencils, which possess an inverse matrix, invertable or regular stencils, to be turned around not invertable stencils as singular stencils designation.

vector cross products (dot product and tensor product)

one has two column vectors <math> v </math> and <math> w</math> the length <math> n </math>, then is the matrix product <math> v \ cdot w </math> does not define, but the two products <math> v^T \ w /math <cdot> and <math> v \ cdot w^T </math> exist.

The first product is one <math> 1 \ times 1 </math> - matrix, asNumber is interpreted, it becomes the canonical dot product of <math> v </math> and <math> w </math> called and with <math> \ langle v, w \ rangle </math> designated.

<math>

\ begin {pmatrix} 1 \ \ 2 \ \ 3 \ end {pmatrix} ^T \ cdot \ begin {pmatrix} to -2 \ \ -1 \ \ 1 \ end {pmatrix} = 1 \ cdot (- 2) + to 2\ cdot (- 1) + 3 \ 1 = -1 cdot

< /math>

The second product is one <math> n \ times n </math> - matrix and is called the dyadische product or tensor product of <math> v </math> and <math> w </math>.

<math>

\ begin {pmatrix} 1 \ \ 2 \ \ 3 \ end {pmatrix} \ cdot \ begin {pmatrix} to -2 \ \-1 \ \ 1 \ end {pmatrix} ^T = \ begin {pmatrix} to 1 \ cdot (- 2) & 1 \ cdot (- 1) & 1 \ cdot 1 \ \ 2 \ cdot (- 2) & 2 \ cdot (- 1) & 2 \ 1 \ \ 3 cdot \ cdot (- 2) & 3 \ cdot (- 1) & 3 \ 1 cdot \ ends {pmatrix} = \ begin {pmatrix} to -2 & -1 & 1 \ \ -4 & -2 & 2 \ \ -6 & -3 & 3\ end {pmatrix}

< to /math>

the transponierte matrix

the Transponierte of the matrix <math> A = (a_ {ij}) </math> of the format <math> m \ times n </math> is the matrix <math> A^T = (a_ {ji}) </math> of the format <math> n \ times m </math>, i.e. too

< math>

A= \ begin {pmatrix} a_ {11}& \ dots &a_ {1n} \ \ \ vdots & \ ddots & \ vdots \ \ a_ {m1} & \ dots &a_ {mn} \ end {pmatrix}

< to /math> is the Transponierte

< math>

 A^ {T} =\ begin {pmatrix}  a_ {11} & \ dots &a_ {m1} \ \  \ vdots & \ ddots & \ vdots \ \  a_ {1n} & \ dots &a_ {mn}\ end {pmatrix}

< to /math>

One writes thus the first line as the first columnand the second line as the second column etc. The matrix is reflected in such a way to say to their main diagonals (11, 22, 33,…, mn).

Example:

<math>

\ begin {pmatrix} 1 & 8 & -3 \ \ 4 & -2 & 5 \ end {pmatrix} ^T = \ begin {pmatrix} to 1 & 4 \ \ 8 & -2\ \ -3 & 5 \ end {pmatrix}

< to /math>

It applies additionally:

<math> (A \ cdot B) ^T=B^T \ A^T /math< cdot>

applications

connection with linear illustrations

the special at stencils over a ring <math> K< /math> the connection is to linear illustrations. To each matrix <math> A \ in K^ {m \ times n}< /math>a linear illustration with definition range leaves itself <math> to K^n< /math> (Quantity of the column vectors) and range of values< math> K^m< /math> define, by one each column vector <math> u \ in K^n< /math> on <math> A \ u cdot \ in K^m< /math> illustrates; and each linear illustration with this definition and range of values corresponds in this way exactly to one <math> m \ times n< /math> - matrix. ThisOne calls connection also (canonical) isomorphicism; it places m /math <with> given< math> K </math>,< math>, <math> n< /math> a Bijektion between the quantity of the stencils and the quantity of the linear illustrations. The stencil product goes here over into the composition (Hintereinanderausführung) of linear illustrations. Because the clasping with thatHintereinanderausführung of three linear illustrations no role plays, applies this then also to the matrix multiplication, it is thus associative.

Is <math> K< /math> even a body, one knows arbitrary finite-dimensional math K /math <instead of>< the column vector spaces> - vector spaces <math> V< /math> and <math> W< /math> (the dimension <math> n< /math> and/or. <math> m< /math>) regard. These are afterChoice of Basen< math> v= (v_1, \ ldots, v_n)< /math> by <math> V< /math> and <math> w= (w_1, \ ldots, w_m)< /math> of <math> W< /math> too <math> K^n< /math> and/or. <math> K^m< /math> isomorphic, because to any vector <math> u \ in V< /math> a clear dismantling in basis vectors <math> u = \ sum_ {i=1} ^n \ alpha_i v_i< /math> existed, and the body elements occurring therein <math> \ alpha_i< /math> the coordinate vector <math> {} _vu= \ begin {pmatrix} \ to alpha_1 \ \ \ vdots \ \ \ alpha_n \ end {pmatrix}\ in K^n< /math> form. However the coordinate vector hangs v /math of <the used> basis< math> off, therefore it also in the designation <math> {} _vu< /math> occurs. (For <math> W< /math> similar applies.)

a linear illustration <math> for f \ colon V \ tons of W< /math> with<math> f (v_i) = \ sum_ {j=1} ^m a_ {ji} w_j< /math> is then completely fixed by the matrix

< math> {} _wf_v = \ begin {pmatrix}a_ {to 11} & \ ldots & a_ {1n} \ \

\ vdots & \ ddots & \ vdots \ \ a_ {m1} & \ ldots & a_ {mn} \ end {pmatrix} \ in K^ {m \ times n},< /math> because for the picture of the above mentioned. Vector <math> u< /math> math

< f> (u) applies = \ sum_ {for j=1} ^m \ sum_ {i=1} ^n a_ {ji} \ alpha_i w_j,< /math>

thus <math> {} _wf (u) = {} _wf_v \ {} _vu /math< cdot> (“Coordinate vector = matrix times coordinate vector”).(The matrix <math> {} _wf_v< /math> v /math hangs of <the used> Basen< math> and <math> w< /math> off; with the multiplication the basis <“>is away-shortened<”> math v /math, which stands on the left and on the right from the point of mark, and “outside” the standing basis <math> w< /math> .)

the Hintereinanderausführung of two linear illustrations remains <math> for f \ colon V \ tons of W< /math> and <math> g \ colonW \ tons of X< /math> (with Basen <math> v< /math>, <math> w< /math> and/or. <math> x< /math>) corresponds thereby to the matrix multiplication, thus <math> {} _x (g \ circ f) _v = {} _xg_w \ cdot {} _wf_v< /math> (the basis becomes math w </math> also< here> “away-shortened”).

Thus is the quantity of the linear illustrations of <math> V< /math> after <math> W< /math> again isomorphic too <math> K^ {m \ times n}< /math>. The isomorphicism <math> f\ mapsto {} _wf_v< /math> however v /math hangs of <the selected> Basen< math> and <math> w< /math> off and is not canonical therefore: With choice of another basis <math> v'< /math> for <math> V< /math> and/or. <math> w'< /math> for <math> W< /math> the same linear illustration another matrix is assigned, those from the old by multiplication ofright and/or. left with one only of the Basen involved dependent invertable <math> m \ times m< /math> - and/or. <math> n \ times n< /math> - matrix (so-called. Basis change matrix) develops. That follows /math by twice application of the multiplication rule from the previous paragraph, <i.e.> math {} _ {w'} f_ {v'} = {} _ {w'} e^W_w \ cdot {} _wf_v \ cdot {} _ve^V_ {<v'}> (“Matrix = basis change matrix timesMatrix times basis change matrix "). The identity illustrations form <math> e^V< for /math> and <math> e^W< /math> each vector out <math> V< /math> and/or. <math> W< /math> on itself off.

If a characteristic remains unaffected by stencils by such basis changes, then it makes for sense to award also basis-independently this characteristic to the appropriate linear illustration.

In connection withStencils often arising terms are rank and the determinant of a matrix. Rank is (if <math> K< /math> a body is basis independent) in the aforementioned sense and one can thus of rank also with linear illustrations speak. The determinant is defined only for square stencils, those thatCase <math> V=W< /math> correspond; it remains unchanged, if the same Basiswechel is accomplished in the definition and range of values, whereby both basis change stencils are to each other inverse: <math> {} _ {v'} f_ {v'} = ({} _ve^V_ {v'}) ^ {- 1} \ cdot {} _vf_v \ cdot {} _ve^V_ {v'}< /math>. In this sense thus also the determinant is basis independent.

transforming stencil equations

special intothe multivariate procedures become frequent proof, derivations etc. accomplished in the stencil calculation.

Equations are transformed in principle as algebraic equations, whereby however the Nichtkommutativität of the matrix multiplication as well as the existence must be considered by zero-divisors.

Example: Linear set of equations as simple shaping

searched is the solution vector <math> x</math> a linear set of equations

< math> A \ cdot x=b< /math>

with <math> A </math> as <math> n \ times n </math> - coefficient matrix. One extended from left

< math> A^ {- 1} \ cdot A \ cdot x=A^ {- 1} \ cdot b \ Leftrightarrow E \ cdot x=A^ {- 1} \ cdot b </math>

and the solution receives

< math> x=A^ {- 1} \ cdot b </math>.

See also further applications.

special stencils

characteristics of Endomorphismen

the following characteristics of square stencils correspond to characteristics of Endomorphismen, which are represented by them.

Orthogonale stencils

a real matrix <math> A< /math> is orthogonal, if the associated linear illustration receives the standard dot product, i.e.if

<math> \ langle Av, Aw \ rangle = \ langle v, w \ rangle< /math>

applies. This condition is equivalent to the fact that <math> A< /math> the equation

<math> A^ {- 1} = A^T< /math>

and/or.

<math> A \, A^T = E< /math>

fulfilled.

These stencils represent reflections, turns and Drehspiegelungen.

Unitary stencils

you are the complex counterpart to the orthogonalen stencils.A complex matrix <math> A< /math> is unitarily, if the associated linear illustration the standard dot product receives along ash, i.e. if

<math> \ langle Av, Aw \ rangle = \ langle v, w \ rangle< /math>

applies. This condition is equivalent to the fact that <math> A< /math> the equation

<math> A^ {- 1} = A^*< /math>

fulfilled; math <A^*> /math< designates> those conjugate transponierte matrix too <math> A< /math>.

Seizesone <math> the n< /math> - dimensional complex vector space as <math> 2n< /math> - dimensional real vector space up, then corresponds the unitary stencils exactly to those orthogonalen stencils, those with the multiplication with <math> \ mathrm i< /math> exchange.

Projection stencils

a matrix is a projection matrix, if

<math> A = A^2< /math>

i.e. the matrix is idempotent, which means,that the repeated application of a projection matrix to a vector leaves the result unchanged. A idempotente matrix does not have a full rank, it is, it is the unit matrix.

Example: It is <math> X< /math> one (mxn) - matrix. Then is (mxm) - the matrix

<math> A = X \, (X^TX) ^ {- 1} X^T< /math>

idempotent. ThisMatrix is used for example in the method of the smallest squares.

Geometrically projection stencils correspond to the Parallelprojektion along the zero-area of the matrix.

Nile-potent stencils

a matrix <math> N< /math> means nil potent, if a power <math> N^k< /math> (and concomitantly all higher powers) the zero-matrix results in.

characteristics ofBilinearformen

in the following are listed to characteristics of stencils, the characteristics of the associated Bilinearform

<math> (v, w) \ mapsto v^T A w< /math>

correspond. Nevertheless these characteristics can possess their own meaning also for the represented Endomorphismen.

Symmetrical stencils

a matrix <math> A </math> means symmetrical, if it equal their transponiertenMatrix is:

<math> A^T = A< /math>

Descriptive spoken the entries of symmetrical stencils are symmetrical to main diagonals.

Example:

<math>

\ begin {pmatrix} 1 & 2 & 3 \ \ 2 & 4 & 5 \ \ 3 & 5 & 6 \ end {pmatrix} ^T = \ begin {pmatrix} to 1 & 2 & 3 \ \ 2& 4 & 5 \ \ 3 & 5 & 6 \ end {pmatrix}

< to /math>

Symmetrical stencils correspond on the one hand to symmetrical Bilinearformen:

<math> v^T A w=w^T A v,< /math>

on the other hand the adjoints linear illustrations:

<math> \ langle Av, w \ rangle = \ langle v, Aw \ rangle. </math>

Hermite stencils

Hermite stencils are the complex analogue of the symmetrical stencils. Itcorrespond to the along ashes Sesquilinearformen and the adjoints Endomorphismen.

A matrix <math> A \ in \ mathbb C^ {n \ times n} </math> is selbstadjungiert hermitesch or, if applies:

<math> A = A^*. </math>

Inclined-symmetrical stencils

a matrix <math> A </math> means inclined symmetrical, if applies:

<math> - A^T = A.< /math>

Example:

<math>

\ begin {pmatrix} 0 & 1 &2 \ \ -1 & 0 & 3 \ \ -2 & -3 & 0 \ end {pmatrix}

< to /math>

Inclined-symmetrical stencils correspond to antisymmetric Bilinearformen:

<math> v^T \ A cdot \ cdot w = - w^T \ A cdot \ cdot v< /math>

and anti- adjoints Endomorophismen:

<math> \ langle Av, w \ rangle = - \ langle v, Aw \ rangle. </math>

Positively definite stencils

a real matrix is positively definite, ifthe associated Bilinearform positively definitely is, i.e. if for all vectors <math> v \ ne0< /math> applies:

<math> v^T \ A cdot \ cdot v > 0< /math>.

Positively definite stencils define generalized dot products. If the Biliniearform is more largely alike to zero, the matrix is called positively semidefinit, similarly can a matrix negatively definite and/or semidefinit to be called,if the above Bilinearform is ever smaller and/or smaller alike to zero. Stencils none of these characteristics fulfill, are called indefinit.

and contains

adjoint matrix conjugated further

constructions a matrix complex numbers, receives one the conjugated matrix, by one its components by those conjugates complex elements replaced. The adjoint matrix (hermitesch matrix also conjugated) of a matrix <math> A </math> becomes with <math> A^*< /math> designated and corresponds to the transponierten matrix, with which additionally all elements are complex conjugated. Sometimes also the complementary matrix becomes <math> A^ \ dagger< /math> adjoint calls.

Adjunkte orComplementary matrix

the complementary matrix <math> A^ \ dagger< /math> a square matrix <math> A </math> consists of their subdeterminants, whereby a subdeterminant also Minor is called. For the determination of the subdeterminants <math> \ December (A_ {ij}) </math> become <math> i </math> - width unit line and <math> j </math> - width unit column of<math> A </math> painted. From the resulting <math> (n-1) \ times (n-1) </math> - matrix becomes then the determinant <math> \ December (A_ {ij}) </math> computed. The complementary matrix has then the entries <math> (- 1) ^ {i+j} \ December (A_ {ij}) </math>. Sometimes this matrix is called also matrix of the Kofaktoren.

One uses the complementary matrixfor example to the computation of the inverse ones of a matrix <math> A </math>, because after the Laplace expansion theorem applies

<math> A^T \ cdot A^ \ dagger= \ December (A) \ cdot E_n< /math>.

Verallgemeinerungen

one can also stencils with infinitely many columns or lines regard. One can add these still. Multiply around it howeverto be able, one must place there additional conditions against its components (the arising sums infinite rows are and not to converge would have).

Similarly to the stencils if mathematical structures with more than two indices are defined, then one calls these tensors.