Lutheranism

In Linear algebra, the comatrice of a square Matrice has is a matrix introduced by a generalization of the calculation of the reverse of has . It has a considerable importance for the study of the determinants. Its coefficients are called cofacteurs has , and they make it possible to study the variations of the function determinant.

The comatrice is also called matrix of the cofacteurs , or, assistant matrix (for example in the Maple software).

Stamp having a variable coefficient

The determinant for the matrices is naturally definite like a function on N vectors columns of the matrix. It is however legitimate to also consider it as a function which with the n² coefficients of the matrix associates a scalar.

When one only freezes all the coefficients of the matrix except for one, the determinant is a Fonction closely connected variable coefficient. The expression of this function closely connected is simple to obtain like particular case of the property of N - linearity; it utilizes a determinant of size n-1 , called cofactor of the variable coefficient.

These considerations make it possible to establish a formula of recurrence bringing back the calculation of a determinant of size N , with that of N determining of size n-1 : it is the formula of Laplace .

Cofactor

Either has a square matrix of size N . One observes the effect of a modification of one of the coefficients of the matrix, all things being equal. For that one thus chooses two indices I for the line and J for the column, and one notes has (X) the matrix whose coefficients are the same ones as those of has , except the term of index I, J which is worth a_{i, j}+x . One writes the formula of linearity for the J - ème column

a_ {i-1,1} & \ dowries & a_ {i-1, j-1} & 0&a_ {i-1, j+1} & \ dowries & a_ {i-1, N} \ \ a_ {I, 1} & \ dowries & a_ {I, j-1} & 1&a_ {I, j+1} & \ dowries & a_ {I, N} \ \ a_ {i+1,1} & \ dowries & a_ {i+1, j-1} & 0&a_ {i+1, j+1} & \ dowries & a_ {i+1, N} \ \ \ vdots & & \ vdots & \ vdots & \ vdots& & \ vdots \ \ a_ {N, 1} & \ dowries & a_ {N, j-1} & 0&a_ {N, j+1} & \ dowries & a_ {N, N} \ end {vmatrix} = \ det A+x {\ rm Cof} _ {I, J}

The determinant noted Cof_{i, j} is called cofactor of index I, J of the matrix has . He admits following interpretations

• to increase X the coefficient of index I, J of the matrix (all things being equal) amounts increasing the determinant of X time the cofactor corresponding
• the cofactor is the derivative of the determinant of the matrix has (X)

In practice, one calculates the cofacteurs in the following way: one calls M (I; J) the determinant of the submatrix deduced from M while having removed line I and the column J (one speaks about minor for such a determinant). Then the cofactor is (- 1) ^i+j time M (I; J).

a_ {i-1,1} & \ dowries & a_ {i-1, j-1} & a_ {i-1, j+1} & \ dowries & a_ {i-1, N} \ \ a_ {i+1,1} & \ dowries & a_ {i+1, j-1} & a_ {i+1, j+1} & \ dowries & a_ {i+1, N} \ \ \ vdots & & \ vdots & \ vdots && \ vdots \ \ a_ {N, 1} & \ dowries & a_ {N, j-1} & a_ {N, j+1} & \ dowries & a_ {N, N} \ end {vmatrix}

Formulas of Laplace

If n>1 and have of size N then one is a square matrix can calculate his determinant according to the coefficients of only one column and the corresponding cofacteurs. This formula, known as formula of Laplace, thus makes it possible to bring back the calculation of the determinant to N calculations of determinants of size n-1 .

• Formule of development compared to the column J

$\backslash \; det\; \{has\}\; =\; \backslash \; sum\_\; \{i=1\}\; ^\; \{N\}\; a\_\; \{I;\; J\}\; \{\backslash \; rm\; Cof\}\; \_\; \{I,\; J\}$

• One can also give a formula of development compared to the line I

$\backslash \; det\; \{has\}\; =\; \backslash \; sum\_\; \{j=1\}\; ^\; \{N\}\; a\_\; \{I;\; J\}\; \{\backslash \; rm\; Cof\}\; \_\; \{I,\; J\}$

Generalization

One introduces the Comatrice of has , matrix made up of the cofacteurs of has . One can generalize the formulas of development of the determinant compared to the lines or columns

$A\; \backslash \; times\; \{\}\; ^t\; =\; \{\}\; ^t\; \backslash \; times\; has\; =\; \backslash \; det\; \{has\}\; \backslash \; times\; I\_n$

The matrix transposed of the comatrice is called complementary matrix has . In particular if has is invertible, the reverse of has is a multiple of the complementary matrix. What wants to say that one obtained a formula for the reverse, not requiring that calculations of determinants

$A^\; \{-\; 1\}\; =\; \backslash \; frac1\; \{\backslash \; det\; has\}\; \backslash ,\; \{\}\; ^t$

This formula is still valid if the matrices are with coefficients in a ring has . It is used to show that M is invertible as a matrix with coefficients in has if and only if det ( M ) is invertible as element of has .

It is of an interest limited to calculate opposite of matrices explicitly; in practice it is too heavy as soon as n=4 and the more elementary method containing elementary operations on the lines (inversion by pivot of Gauss) is more effective, as well for the man as for the machine.

Properties of the comatrice

We have

COM ( I _N ) = I _N

and

for all matrices of order N M and NR , COM ( MN ) = COM ( NR ) COM ( M )

The comatrice is also compatible with the transposition:

COM (^t M ) = ^t (COM ( M )).

moreover,

det (COM ( M )) = det ( M ) ^{N -1}.

If p ( T ) = det ( M - Ti _N ) is the Polynôme characteristic of M and that Q is the polynomial defined by Q ( T ) = ( p (0) - p ( T ))/ T , then

^tcom ( M ) = Q ( M ).

The comatrice appears in the formula of the Dérivée from a determinant.

For $A\; \backslash \; in\; M\_\; \{N\}\; (K)$:

• if has is of row N (i.e. has invertible), COM (A) too. There is then $Com\; (A)=det\; (A)~^\; \{T\}\; A^\; \{-\; 1\}$ and $Com\; (A)^\; \{-\; 1\}\; =\; \backslash \; frac\; \{1\}\; \{det\; (A)\}\; ~^\; \{T\}\; A$.
• if has is of row n-1, COM (A) is of row 1.
• if has is of row to more N2, COM (A)=0.

If $n\; \backslash \; geq\; 3$ and $A\; \backslash \; in\; M\_\; \{N\}\; (K)$, $Com\; (COM\; (A))\; =det\; (A)^\; \{N2\}\; \backslash ,\; A$ (and is thus null if, and only if, has is not invertible). If n=2, one has COM (COM (A)) =A for any matrix has (what one can include in the preceding formula with convention $x^\; \{0\}\; =1$ for all $x\; \backslash \; in\; K$, including for x=0).

If $n\; \backslash \; geq\; 3$, the matrices $A\; \backslash \; in\; M\_\; \{N\}\; (\backslash \; mathbb\; \{R\})$ such as A=Com (A) are the null matrix and the orthogonal special matrices. If n=2, they are the multiple matrices of the orthogonal special matrices.

Variations of the determining function

The formula of Leibniz watch that the determinant of a matrix has expresses like nap and product of components of has . It is thus not astonishing that the determinant has good properties of regularity. It is supposed here that K is the body of realities.

Determinant dependant on a parameter

If $t\; \backslash \; mapsto\; has\; (T)$ is a function of class $\backslash \; mathcal\; C^k$ with values in the square matrices of order N , then $t\; \backslash \; mapsto\; \backslash \; det\; has\; (T)$ is also of class $\backslash \; mathcal\; C^k$.

The formula of derivation is obtained while utilizing the columns of has

$\backslash \; frac\; \backslash \; left\; (\backslash \; det\; (A\_1\; (T),\; \backslash \; dowries,\; A\_n\; (T))\; \backslash \; right)\; =\; \backslash \; sum\_\; \{i=1\}\; ^n\; \backslash \; det\; (A\_1\; (T),\; \backslash \; dowries,\; A\_\; \{i-1\}\; (T),\; A\text{'}\_i\; (T),\; A\_\; \{i+1\}\; (T),\; \backslash \; dowries,\; A\_n\; (T))$

This formula is similar formally to derived from a product of numerical N functions.

The determinant like function on the space of the matrices

• the application which with the matrix has associates its determinant is continuous.

This property has interesting topological consequences: thus the group GL_n ($\backslash \; mathbb\; \{R\}$) is a Ouvert, the sub-group SL_n ($\backslash \; mathbb\; \{R\}$) one is closed.

• This application is in fact differentiable , and even $\backslash \; mathcal\; C^\; \backslash \; infty$

Indeed the calculation of the cofacteurs can be considering precisely like a calculation of partial derivative

$\backslash \; frac\; \{\backslash \; partial\; \backslash \; det\}\; \{\backslash \; partial\; E\_\; \{ij\}\}\; (A)\; =\; \{\backslash \; rm\; Cof\}\; A\_\; \{I,\; J\}$

All these derivative partial being themselves of the determinants, by recurrence the determinant is $\backslash \; mathcal\; C^\; \backslash \; infty$. Moreover one can write the development limited to the order one of the determinant in the vicinity of has

$\backslash \; det\; (A+H)\; =\; \backslash \; det\; has\; +\; \{\backslash \; rm\; tr\}\; (\{\}\; ^t\; \{\backslash \; rm\; COM\}\; (A).H)\; +o\; (\backslash |H\; \backslash |)$

I.e. if one provides M_n ($\backslash \; mathbb\; \{R\}$) with his canonical scalar product, the determining application has as a gradient

$\backslash \; nabla\; \backslash \; det\; (A)\; =\; \{\backslash \; rm\; COM\}\; (A)$

In particular for the case where has is the identity

$\backslash \; det\; (I+H)\; =1\; +\; \{\backslash \; rm\; tr\}\; (H)\; +o\; (\backslash |H\; \backslash |)\; \backslash \; qquad\; \backslash \; nabla\; \backslash \; det\; (I)\; =\; I$

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