1/40 Determinants Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra
About Determinant A scalar function on the set of square matrices A handful of basic properties An expression for matrix inverse An expression for the solution to a system of linear equations A key to eigenvalue problems 2/40 Prof. Chen Determinants 2 / 40
Basic Properties 3/40 Prof. Chen Determinants 3 / 40
Identity Matrix The determinant of an identity matrix is 1. 1 0 I 2 = 0 1 = 1 1 0 0 I 3 = 0 1 0 = 1 0 0 1 4/40 Prof. Chen Determinants 4 / 40
Exchange of Adjacent Rows The determinant of a matrix changes sign if two adjacent rows are exchanged. That is A ai : a i+1 : = A, i 3 4 1 2 = 1 2 3 4 3 4 5 1 2 3 1 2 3 = 3 4 5 1 0 1 1 0 1 5/40 Prof. Chen Determinants 5 / 40
Linearity in Row 1 The determinant of a matrix is linear in row 1. A a1: =αb+βc = α A a1: =b + β A a1: =c 3 7 3 2 = 1 3 3 2 + 2 4 3 2 = 3 0 3 2 + 0 7 3 2 3 7 1 2 3 3 1 4 2 3 2 1 + 3 2 1 = 3 2 1 1 1 2 1 1 2 1 1 2 6/40 Prof. Chen Determinants 6 / 40
Corollary 7/40 Prof. Chen Determinants 7 / 40
Exchange of Rows The determinant of a matrix changes sign if any two rows are exchanged. That is A ai: a j: = A, i j For example 3 4 5 1 0 1 1 0 1 1 2 3 = ( 1) 1 2 3 = ( 1) 2 3 4 5 1 0 1 3 4 5 1 2 3 The corollary follows from that an exchange of non-adjacent rows is equivalent to an odd number of exchanges of adjacent rows. (Why?) 8/40 Prof. Chen Determinants 8 / 40
Permutation Matrix Since a permutation matrix, say P, results from row exchanges of an identity matrix P = ±1 0 1 0 0 0 1 = 1 1 0 0 0 1 0 0 0 0 1 0 = 1 0 0 0 1 1 0 0 0 9/40 Prof. Chen Determinants 9 / 40
10/40 Linearity in Row The determinant of a matrix is linear in any row vector. A ai: =αb+βc = α A ai: =b + β A ai: =c, i a 1: a i: = αb + βc = αb + βc a 1: = α b a 1: β c a 1: = α a 1: a i: = b + β a 1: a i: = c Prof. Chen Determinants 10 / 40
Derived Properties 11/40 Prof. Chen Determinants 11 / 40
Equal Rows The determinant of a matrix with two equal rows is 0. i j a i: = a j: A = 0 Suppose a i: = a j: (i j) Then by row exchange and since A ai: a j: = A A ai: a j: = A and A ai: a j: = A A = A A = 0 12/40 Prof. Chen Determinants 12 / 40
Row Operation The determinant of a matrix is invariant with respect to row operation. A a j: a j: ka i:, i j A A = A A aj: a j: ka i: = A aj: a j: A aj: ka i: = A k A aj: a i: = A 0 = A 13/40 Prof. Chen Determinants 13 / 40
Row of Zeros The determinant of a matrix with a row of zeros is 0. Suppose For any row i a j: = 0 T A = A aj: a j: +a i: = A aj: 0 T +a i: = A aj: a i: = 0 14/40 Prof. Chen Determinants 14 / 40
Examples equal rows row operation row of zeros 1 2 1 2 = 0 1 2 3 4 = 1 2 2 2 1 2 0 0 = 0 15/40 Prof. Chen Determinants 15 / 40
Triangular Matrix The determinant of a lower-triangular matrix is the product of the diagonal elements. If the diagonal elements are non-zero A row operations D A = D = a 11... a nn I n = a 11... a nn Otherwise, let a kk be the first zero on the diagonal. Then 0, i < k (by row operations) a ki = 0, i = k (by assumption) 0, i > k (by lower-triangularity) so row k is 0 T, and A = 0 = a 11... a nn 16/40 Prof. Chen Determinants 16 / 40
Singular Matrix The determinant of a singular matrix is 0. Suppose A is singular. Let A Gaussian elimination U Then U has at least one row of zeros, so U = 0 It follows that A = U = 0 17/40 Prof. Chen Determinants 17 / 40
Matrix Product The determinant of matrix product is the product of matrix determinants AB = A B 18/40 Prof. Chen Determinants 18 / 40
Proof For any B, consider the scalar function defined by f (A) = AB. We B have f (I) = IB B = B B = 1 A = A ai: a i+1: A B = (AB) ai: B a i+1: B A B = AB f (A ) = f (A) a 1: = αc + βd a 1: B = αcb + βdb So f (A) = A. It follows that A a1: =αc+βdb = α A a1: =cb + β A a1: =db f (A a1: =αc+βd) = αf (A a1: =c) + βf (A a1: =d) AB = A B 19/40 Prof. Chen Determinants 19 / 40
Matrix Transpose The determinant does not change with matrix transpose. If A is singular, A T is singular, and A = A T = 0 If A is non-singular, there exists an LDU decomposition PA = LDU. P A = L D U = U T D T L T = U T D T L T P is a permutation matrix so = A T P T = A T P T P T P = I P =±1 P = P T A = A T 20/40 Prof. Chen Determinants 20 / 40
Examples triangular matrix singular matrix 1 0 2 3 = 1 3 [ ] 1 2 1 2 2 4 = 0 2 4 singular matrix product matrix transpose 1 2 4 3 3 4 2 1 = 8 5 20 13 1 3 2 4 = 1 2 3 4 21/40 Prof. Chen Determinants 21 / 40
Computation of Determinant 22/40 Prof. Chen Determinants 22 / 40
Product of Pivots Apart from a sign, the determinant of a non-singular matrix is equal to the product of its pivots. Suppose A is non-singular with an LDU decomposition PA = LDU Taking determinants on both sides, we have P A = L D U = D Since P = ±1, we have A = ± D = ± i d i 23/40 Prof. Chen Determinants 23 / 40
Example 2 1 2 1 2 1 A = 1 2 = L 1 1 2 So the pivots are and 2, 3 2, 4 3,..., n + 1 n ( ) ( ( ) 3 4 n + 1 A = 2... = n + 1 2 3) n 3 2 4 3 n+1 n U 24/40 Prof. Chen Determinants 24 / 40
Sum of Determinants over Permutations The determinant of a matrix of order n n is the sum of n! terms, each being the product of n elements in the matrix not occupying the same row or the same column. A = (α,...,ν) (a 1α... a nν ) P(α,..., ν) where the summation is over all permutations of (1, 2,..., n), and P(α,..., ν) is the permutation matrix with 1 s at (1, α),..., (n, ν) 25/40 Prof. Chen Determinants 25 / 40
The Argument By the linearity property, the determinant of an n n matrix can be expressed as the sum of n n determinants, by treating each row as the combination of n 1-hot vectors. For example, n = 2 a 11 a 12 a 21 a 22 = a 11 0 a 21 0 + a 11 0 0 a 22 + 0 a 12 a 21 0 + 0 a 12 0 a 22 Most of the n n determinants are 0 due to zero columns. Specifically, a ij s must occupy different columns, or the determinant is 0. Keeping only such terms, we have A = (α,...,ν) (a 1α... a nν ) P(α,..., ν) 26/40 Prof. Chen Determinants 26 / 40
Examples a 11 a 12 a 21 a 22 = a 11 0 a 21 a 22 + 0 a 12 a 21 a 22 a = 11 0 a 21 0 + a 11 0 0 a 22 + 0 a 12 a 21 0 + 0 a 12 0 a 22 a = 11 0 0 a 22 + 0 a 12 a 21 0 a 11 a 12 a 13 a 11 0 0 a 11 0 0 0 a 12 0 a 21 a 22 a 23 = 0 a 22 0 + 0 0 a 23 + 0 0 a 23 a 31 a 32 a 33 0 0 a 33 0 a 32 0 a 31 0 0 0 a 12 0 0 0 a 13 0 0 a 13 + a 21 0 0 + a 21 0 0 + 0 a 22 0 0 0 a 33 0 a 32 0 a 31 0 0 27/40 Prof. Chen Determinants 27 / 40
Cofactor Expansion The determinant of a matrix can be expanded along row 1 by A = a 11 c 11 + a 12 c 12 + + a 1n c 1n where c 1j is the cofactor of a 1j given by c 1j = ( 1) 1+j M 1j where M 1j is the matrix resulting from deleting a 1: and a j of A. 28/40 Prof. Chen Determinants 28 / 40
Proof By linearity, A = n K j, where j=1 0... 0 a 1j 0... 0 a K j = 21... a 2(j 1) a 2j a 2(j+1)... a 2n....... a n1... a n(j 1) a nj a n(j+1)... a nn If a 1j = 0 we have K j = 0 = a 1j c 1j. Otherwise, with row operations 0... 0 a 1j 0... 0 a K j = 21... a 2(j 1) 0 a 2(j+1)... a 2n....... a n1... a n(j 1) 0 a n(j+1)... a nn 29/40 Prof. Chen Determinants 29 / 40
With j 1 exchanges of adjacent columns a 1j 0... 0 0... 0 K j = ( 1) j 1 0 a 21... a 2(j 1) a 2(j+1)... a 2n....... 0 a n1... a n(j 1) a n(j+1)... a nn = a 1j ( 1) j 1 M 1j = a 1j ( 1) j+1 M 1j = a 1j c 1j Thus n n A = K j = a 1j c 1j = a 11 c 11 + a 12 c 12 + + a 1n c 1n j=1 j=1 30/40 Prof. Chen Determinants 30 / 40
General Cofactor Expansion By the same argument, cofactor expansion of a determinant can be done along any row or any column. Element a ij of A has cofactor c ij = ( 1) i+j M ij where M ij is the matrix obtained by deleting a i: and a j of A. The cofactor expansion of A along row i is n A = a ij c ij j=1 Along column j, we have n A = a ij c ij i=1 31/40 Prof. Chen Determinants 31 / 40
Examples product of pivots a c b d = 1 0 a 0 c a 1 1 b 0 ad bc a 0 1 a sum of determinants over permutations cofactor expansion a c b = ad bc d ( ) ad bc = a a 2 1 0 0 1 2 1 0 0 1 2 1 = 2 1 0 2( 1)2 1 2 1 0 0 1 2 0 1 2 + 1 1 0 ( 1)( 1)3 0 2 1 0 1 2 = 2(4) 3 = 5 32/40 Prof. Chen Determinants 32 / 40
Applications 33/40 Prof. Chen Determinants 33 / 40
Cofactors and Row/Column Replacement The cofactor c ij of a ij does not depend on row i or column j. Row replacement a 11 a 12 a 13 a 11 a 12 a 13 a 21 a 22 a 23 and a 11 a 12 a 13 a 31 a 32 a 33 a 31 a 32 a 33 have the same cofactors along row 2. Column replacement a 11 a 12 a 13 a 11 a 12 b 1 a 21 a 22 a 23 and a 21 a 22 b 2 a 31 a 32 a 33 a 31 a 32 b 3 have the same cofactors along column 3. 34/40 Prof. Chen Determinants 34 / 40
Matrix Inverse with Cofactors Suppose A is non-singular. Then A 1 = A 1 C T where c 11 c 12... c 1n c C = 21 c 22... c 2n...... c n1 c n2... c nn is the cofactor matrix of A with the cofactors c ij as elements. 35/40 Prof. Chen Determinants 35 / 40
Proof We show AC T = A I (from which A 1 = A 1 C T follows). Now AC T ij = k a ik c T kj = k a ik c jk = a i1 c j1 + + a in c jn = A aj: a i: as the sum is the cofactor expansion of A aj: a i: along row j. Since 0, i j A aj: a i: = A, i = j we have AC T ij = A δ ij, so AC T = A I 36/40 Prof. Chen Determinants 36 / 40
Example [ ] 1 a b = c d 1 ad bc [ d ] b c a 1 1 1 1 1 1 0 0 1 1 = 0 1 1 0 0 1 0 0 1 37/40 Prof. Chen Determinants 37 / 40
Cramer s Rule The solution of a non-singular system of linear equations Ax = b is x j = A a j b A a 1... a j 1 b a j+1... a n =, j = 1,..., n a 1... a j 1 a j a j+1... a n 38/40 Prof. Chen Determinants 38 / 40
Proof Using previous result, we have Let x = A 1 b = A 1 C T b n n x j = A 1 C T b j = A 1 cji T b i = A 1 b i c ij i=1 i=1 A = A aj b be the matrix formed by replacing a j of A with b. As is shown earlier, A and A have the same cofactors along column j. The sum n b i c ij is the cofactor expansion of A along column j (with a ij = b i ), so x j = A 1 A = A a j b A i=1 39/40 Prof. Chen Determinants 39 / 40
Example x 1 + 3x 2 = 0 2x 1 + 4x 2 = 6 By Cramer s rule, the solution is 0 3 6 4 x 1 = 1 3 = 9, x 2 = 2 4 1 0 2 6 1 3 2 4 = 3 40/40 Prof. Chen Determinants 40 / 40