nLab Laplace expansion

Contents

Idea

The determinant of a square matrix with commutative entries can be expanded into a sum of signed products of all maximal minors in a chosen proper subset of the sets of rows (or columns) with the complementary minors.

Analogous formulas hold for quantum determinants.

Laplace expansion identity

Let AA be a quadratic n×nn\times n-matrix with entries in a commutative ring RR and K,L{1,,n}K,L\subset \{1,\ldots,n\}, K^={1,,n}\K\hat{K} = \{1,\ldots,n\}\backslash K and card(K)=card(L)=mcard(K) =card(L) = m. Denote by A J IA^I_J the submatrix of AA with rows in II and columns in JJ for I,J{1,,n}I,J\subset\{1,\ldots,n\}. Then

(1) l(LL^)δ K Ldet(T)= card(J)=m(1) l(JJ^)det(A K J)det(T L^ J^) (1) l(LL^)δ L Kdet(T)= card(J)=m(1) l(JJ^)det(T J K)det(T J^ L^) \begin{array}{l} (-1)^{l(L\star \hat{L})} \delta^L_K\, det(T) = \sum_{card(J) = m} (-1)^{l(J\star \hat{J})} \, det(A^J_K) det(T^{\hat{J}}_{\hat{L}})\\ (-1)^{l(L\star \hat{L})}\delta^K_L\, det(T) = \sum_{card(J) = m} (-1)^{l(J\star \hat{J})} \, det(T^K_J) det(T^{\hat{L}}_{\hat{J}})\\ \end{array}

where ll is the length function of a sequence considered as a permutation of {1,,n}\{1,\ldots,n\} and \star denotes the concatenation and δ\delta is the Kronecker for multiindices.

Notice that if m=n1m = n-1 then J={j}J = \{j\} and l(JJ^)=l(j,1,,j1,j+1,n)=j1l(J\star \hat{J}) = l(j,1,\ldots,j-1,j+1,\ldots n) = j-1.

category: algebra

Last revised on June 5, 2024 at 12:56:57. See the history of this page for a list of all contributions to it.