lgebra Linear e Aplicaes DETERMINANTS Idea older than matrices - PowerPoint PPT Presentation

Álgebra Linear e Aplicações

DETERMINANTS

Idea older than matrices • Dates back at least to the 1650s • Seki Kowa • Leibinitz • Popular between 1750 and 1900 • Major tool to analyze and solve linear systems • Gave way to Cayley’s matrix algebra • Determinants still important in theory • Not so much for practical applications

Permutations • A permutation p = ( p 1 , p 2 , p 3 , . . . , p n ) of the numbers is a (1 , 2 , 3 , . . . , n ) rearrangement • There are of them n ! = n ( n − 1)( n − 2) · · · 1 • Can be restored to natural order in many ways, by different numbers of inversions (1 , 4 , 3 , 2) (1 , 2 , 3 , 4) (1 , 4 , 2 , 3) (1 , 2 , 4 , 3) (1 , 2 , 3 , 4)

Parity of inversions • The parity of the number of inversions needed to restore a permutation is unique • Different ways to prove • Decompose into adjacent inversions • Use quotient of polynomials • So define ( even # of inversions 1 σ ( p ) = odd # of inversions − 1

Definition of determinant • For an n × n matrix , the A = [ a ij ] determinant of A is X det( A ) = | A | = σ ( p ) a 1 p 1 a 2 p 2 · · · a np n p • Sum is over all n ! permutations of p = ( p 1 , p 2 , p 3 , . . . , p n ) (1 , 2 , 3 , . . . , n ) • Each term contains one element from one column and one row • No determinant for non-square matrices

Properties #1 • Determinant of triangular matrices is product of entries in the diagonal det( A ) = det( A T ) and • det( A ∗ ) = det( A ) • Effects of row operations from A to B • Exchange rows i and j det( B ) = − det( A ) • Multiply row i by det( B ) = α det( A ) α • Add times row i to row j det( B ) = det( A ) α • Determinants of corresponding elementary matrices are, respectively, , , and − 1 1 α

Properties #2 • For an elementary matrix P , we have det( PA ) = det( P ) det( A ) • Matrix A is singular if and only if det( A ) = 0 • det( AB ) = det( A ) det( B ) � � A B • � � � = det( A ) det( C ) � � 0 C � is the size of the largest non-zero minor • rank ( A ) • A minor of A is the determinant of a submatrix

Volumes and determinants • Let have independent columns. A ∈ R m × n The volume of the n -dimensional parallelepiped formed by the columns of A is ⇤ 1 det( A T A ) ⇥ V n = 2 • If A is square, this reduces to � det( A ) � � V n = �

Volumes and QR #1 • Let contain n vectors in R m { x 1 , x 2 , . . . , x n } • What is the volume of the parallelepiped? V 2 = k x 1 k 2 k ( I � P 2 ) x 2 k 2 = α 1 α 2 V 3 = k x 1 k 2 k ( I � P 2 ) x 2 k 2 k ( I � P 3 ) x 3 k 2 = α 1 α 2 α 3 V n = k x 1 k 2 k ( I � P 2 ) x 2 k 2 · · · k ( I � P n ) x n k 2 = α 1 α 2 · · · α n

Volumes and QR #2 V n = k x 1 k 2 k ( I � P 2 ) x 2 k 2 · · · k ( I � P n ) x n k 2 = α 1 α 2 · · · α n • This is exactly what orthogonal reduction does! q 1 T x 2 q 1 T x n 2 3 α 1 · · · q 2 T x n 0 α 2 · · · 6 7 6 7 . . ... . . 6 7 . . 6 7 6 7 0 0 ⇥ x 1 ⇤ ⇥ q 1 ⇤ α n = 6 7 · · · x 2 x n q 2 q m · · · · · · 6 7 6 7 0 0 0 · · · 6 7 6 7 . . . ... . . . 6 7 . . . 4 5 0 0 0 · · · i − 1 X q T x i = α i q i + k x i k =1

Volumes and determinants • Proof det( A T A ) = det( R T Q T QR ) = det( R T R )  S � = det( S T S ) with R = 0 = det( S ) 2 = ( α 1 α 2 · · · α n ) 2 = V 2 n

Rank-One Updates • If is non-singular, and c , d ∈ R n × 1 A ∈ R n × n det( I + cd T ) = 1 + d T c • det( A + cd T ) = det( A )(1 + d T A − 1 c ) • • Proof  I �  I + cd T �  �  I � 0 c I 0 c = 1 + d T c d T − d T 0 0 1 1 1 A + cd T = A ( I + A − 1 cd T )

Cramer’s rule • In a non-singular system , we have A n × n x = b x i = det( A i ) det( A ) where ⇥ ⇤ A ∗ 1 | · · · | A ∗ i − 1 | b | · · · | A ∗ n A i = • Proof A i = A + ( b − A ∗ i ) e T i A + ( b − A ∗ i ) e T � � det( A i ) = det i 1 + e T � i A − 1 ( b − A ∗ i ) � = det( A ) 1 + e T � � = det( A ) x i = det( A ) i ( x − e i )

Cofactors and expansion • The cofactor of associated to is ( i, j ) A n × n ˚ A ij = ( − 1) i + j M ij where M ij is the minor • The matrix of cofactors is denoted ˚ A • The determinant can be expanded by cofactors i =1 a ij ˚ det( A ) = P n about column j • A ij j =1 a ij ˚ about row i det( A ) = P n • A ij

Determinant formula for inverse adj( A ) = ˚ • The adjugate of is A T A n × n • The transpose of the coffactor matrix • Some older texts call this the adjoint matrix • If A is non-singular, then ˚ A T det( A ) = adj( A ) A − 1 = det( A ) • Proof x i = det( A i ) [ A − 1 ] ij = x i Ax = e j det( A ) det( A i ) = ˚ ⇥ ⇤ A ∗ 1 | · · · | A ∗ i − 1 | e j | · · · | A ∗ n A i = A ji