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
• f the numbers is a

rearrangement

• There are of them
• Can be restored to natural order in many ways,

by different numbers of inversions p = (p1, p2, p3, . . . , pn) (1, 2, 3, . . . , n) n! = n(n − 1)(n − 2) · · · 1

(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

σ(p) = ( 1 −1 even # of inversions

• dd # of inversions

Definition of determinant

• For an n × n matrix , the

determinant of A is

• Sum is over all n! permutations
• f
• Each term contains one element from one

column and one row

• No determinant for non-square matrices

A = [aij]

det(A) = |A| = X

p

σ(p) a1p1a2p2 · · · anpn

p = (p1, p2, p3, . . . , pn) (1, 2, 3, . . . , n)

Properties #1

• Determinant of triangular matrices is product
• f entries in the diagonal
• and
• Effects of row operations from A to B
• Exchange rows i and j
• Multiply row i by
• Add times row i to row j
• Determinants of corresponding elementary

matrices are, respectively, , , and

det(B) = − det(A) α det(B) = α det(A) α det(B) = det(A)

det(A) = det(AT ) det(A∗) = det(A) α −1 1

Properties #2

• For an elementary matrix P, we have
• Matrix A is singular if and only if
• is the size of the largest non-zero minor
• A minor of A is the determinant of a submatrix

det(PA) = det(P) det(A)

det(A) = 0

det(AB) = det(A) det(B) rank(A)

• A

B C

• = det(A) det(C)

Volumes and determinants

• Let have independent columns.

The volume of the n-dimensional parallelepiped formed by the columns of A is

• If A is square, this reduces to

A ∈ Rm×n Vn = ⇥ det(AT A) ⇤ 1

2

Vn =

• det(A)

Volumes and QR #1

• Let contain n vectors in Rm
• What is the volume of the parallelepiped?

{x1, x2, . . . , xn}

V3 = kx1k2 k(I P2)x2k2 k(I P3)x3k2 = α1α2α3 V2 = kx1k2 k(I P2)x2k2 = α1α2 Vn = kx1k2 k(I P2)x2k2 · · · k(I Pn)xnk2 = α1α2 · · · αn

Volumes and QR #2

• This is exactly what orthogonal reduction does!

⇥x1 x2 · · · xn ⇤ = ⇥q1 q2 · · · qm ⇤ 2 6 6 6 6 6 6 6 6 6 6 6 4 α1 q1T x2 · · · q1T xn α2 · · · q2T xn . . . ... . . . · · · αn · · · . . . . . . ... . . . · · · 3 7 7 7 7 7 7 7 7 7 7 7 5

Vn = kx1k2 k(I P2)x2k2 · · · k(I Pn)xnk2 = α1α2 · · · αn

xi = αiqi +

i−1

X

k=1

qT

k xi

Volumes and determinants

• Proof

det(AT A) = det(RT QT QR) = det(RT R) = det(S)2 = (α1α2 · · · αn)2 = V 2

n

R = S

• = det(ST S)

with

Rank-One Updates

• If is non-singular, and
• Proof

A ∈ Rn×n c, d ∈ Rn×1

det(I + cdT ) = 1 + dT c det(A + cdT ) = det(A)(1 + dT A−1c)  I dT 1  I + cdT c 1  I −dT 1

• =

I c 1 + dT c

• A + cdT = A(I + A−1cdT )

Cramer’s rule

• In a non-singular system , we have

where

• Proof

An×nx = b xi = det(Ai) det(A) Ai = ⇥ A∗1 | · · · | A∗i−1 | b | · · · | A∗n ⇤ Ai = A + (b − A∗i)eT

i

det(Ai) = det

• A + (b − A∗i)eT

i

• = det(A)
• 1 + eT

i A−1(b − A∗i)

• = det(A)
• 1 + eT

i (x − ei)

• = det(A) xi

Cofactors and expansion

• The cofactor of associated to is

where Mij is the minor

• The matrix of cofactors is denoted
• The determinant can be expanded by cofactors
• about column j
• about row i

An×n (i, j) ˚ Aij = (−1)i+jMij ˚ A

det(A) = Pn

i=1 aij ˚

Aij det(A) = Pn

j=1 aij ˚

Aij

Determinant formula for inverse

• The adjugate of is
• The transpose of the coffactor matrix
• Some older texts call this the adjoint matrix
• If A is non-singular, then
• Proof

An×n A−1 = ˚ AT det(A) = adj(A) det(A) adj(A) = ˚ AT

[A−1]ij = xi Ax = ej xi = det(Ai) det(A) Ai = ⇥ A∗1 | · · · | A∗i−1 | ej | · · · | A∗n ⇤ det(Ai) = ˚ Aji