# week9 - 9 Linear algebra Properties of and operations with...

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9 Linear algebra Read: Boas Ch. 3. 9.1 Properties of and operations with matrices M × N matrix with elements A ij A = A 11 A 12 ... A 1 j ... A 1 N A 21 A 22 ... A 2 j ... A 2 N . . . . . . . . . . . . A i 1 A i 2 ... A ij ... A iN . . . . . . . . . . . . A M 1 A M 2 ... A Mj ... A MN (1) Deﬁnitions: Matrices are equal if their elements are equal, A = B A ij = B ij . ( A + B ) ij = A ij + B ij ( kA ) ij = kA ij for k const. ( AB ) ij = N =1 A i‘ B ‘j . Note for multiplication of rectangular matrices, need ( M × N ) · ( N × P ). Matrices need not “commute”. AB not nec. equal to BA . [ A,B ] AB - BA is called “commutator of A and B . If [ A,B ] = 0, A and B commute. For square mats. N × N , det A = | A | = π sgn π A 1 π (1) A 2 π (2) ...A ( N ) , where sum is taken over all permutiations π of the elements { 1 ,...N } . Each term in the sum is a product of N elements, each taken from a diﬀerent row of A and from a diﬀerent column of A , and sgn π . Examples: A 11 A 12 A 21 A 22 = A 11 A 22 - A 12 A 21 , (2) A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 = A 11 A 22 A 33 + A 12 A 23 A 31 + A 13 A 21 A 32 - A 12 A 21 A 33 - A 13 A 22 A 31 - A 11 A 22 A 32 (3) det AB = det A · det B but det( A + B ) 6 = det A + det B . For practice with determinants, see Boas. 1

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Identity matrix I : IA = A A . I ij = δ ij . Inverse of a matrix. A · A - 1 = A - 1 A = I . Transpose of a matrix ( A T ) ij = A ji . Formula for ﬁnding inverse: A - 1 = 1 det A C T , (4) where C is “cofactor matrix”. An element C ij is the determinant of the N - 1 × N - 1 matrix you get when you cross out the row and column (i,j), and multiply by (
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week9 - 9 Linear algebra Properties of and operations with...

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