Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte. Let consider the following set of n simultneous lgebric equtions: x x 2 2 x n n y x x 2 22 2 x 2n n y 2 x x x y n n2 2 nn n n We my simplify to X defined to be: Y ;, X, nd Y re defined s mtrices. These three mtrices re n 2 n 2 22 2n nn X x x 2 x n Y y y 2 y n (D-) where the brcketed rrys re simplified of coefficients nd vribles. Definition of Mtrix mtrix is collection of elements rrnged in rectngulr or squre rry. Severl wys of representing mtrix re s follows: 0 0 0 2 4 2 4 2 4 2,2 (D-2) Mtrix Elements: 2 3 2 22 23 3 32 33
is defined s the element in the i th row nd the j th column of the mtrix (Row st nd column lst). Order of Mtrix: refers to the totl number of rows nd columns of the mtrix. In generl, mtrix with n rows nd m columns is termed "n x m" or "n by m". Squre Mtrix: one tht hs one column nd more thn one row: mx mtrix m >. Row Mtrix: is one tht hs one row nd more thn one column: xn mtrix n >. Digonl Mtrix: is squre mtrix with 0 for ll i j 0 0 0 0 22 0 0 33 (D-3) Unit Mtrix (Identity Mtrix): is digonl mtrix with ll the elements on the min digonl i j equl to. unity mtrix is often designted by either I or U. Unity exmple is I 0 0 0 0 0 0 (D-4) Null Mtrix: is one whose elements re equl to zero for exmple: O 0 0 0 0 0 0 (D-5) Symmetric Mtrix: is squre mtrix tht stisfies the condition 6 5 4 5 0 7 ; 4 7. For exmple: ji
Determinnt of Mtrix: with ech squre mtrix determinnt hving the sme elements nd order. The determinnt of squre mtrix is designted by: Det D Determinnt of the mtrix c b d is define s c b d d bc Consider the mtrix: 0 0 3 2 0 Then: 0 0 3 2 0 3 2 0 2 0 3 0 ( ) 0 0 (0 2) 0(0 2) ( )(0 3) 2 3 2 2 22 23 2 22 3 32 33 3 32 22 33 2 23 3 3 2 32 3 22 3 23 32 2 2 33 Singulr Mtrix: is sid to be singulr if the vlue of its determinnt is zero. Where mtrix is singulr, it usully mens tht not ll the rows or not ll the columns of the mtrix re independent of ech other. Let consider the following set of equtions: 2x 3x x 0 2 3 x x x 0 2 3 x 2x 2x 0 2 3 The third eqution is equl to the sum of the first two equtions. Therefore these three equtions re not completely independent. In mtrix form, these equtions my be represented by X = 0
2 3 Where 2 2 & X x x x 2 3 Determinnt of : 2 3 2 2 0 Therefore the mtrix is singulr. Trnspose of Mtrix: The trnspose of mtrix is defined s the mtrix tht is obtined by interchnging the corresponding rows nd columns in. Let be n nxm mtrix which is represented by. For exmple: nm, Then the trnspose of, denoted by ' is given by ' = trnspose of. mn, 3 0 3 2 2 0 5 5 Skew-Symmetric Mtrix: is squre mtrix tht equls its negtive trnspose, tht is: Some Opertions of Mtrix Trnspose: k k ; where k is sclr B B B B djoint of Mtrix: Let be squre mtrix of order n. The djoint mtrix of denoted by dj( ) is defined s dj( ) cofctor of Det, ; where the nn cofctor of the determinnt of is the determinnt obtined by
omitting the i th row nd the j th column of nd then multiplying by i j. Exmple D-: Determine the djoint mtrix of: c b d The determinnt of is c b d d b dj c Exmple D-2: dj 2 3 2 22 23 3 32 33 22 33 23 32 2 33 3 32 2 23 3 22 2 33 23 3 33 3 3 23 3 2 2 32 22 3 32 2 3 22 2 2 Summtion or Subtrction of Mtrix: Given two m x n mtrices nd B b their sum is nd their difference is B b B b Sclr Multipliction: The sclr product of number k nd mtrix is denoted by k. k k 2 2 k k k k 2 22 2 22
Multipliction of Mtrices: Let be n m x n mtrix nd let B be n n x k mtrix. To find the element in the i th row nd j th column of the product mtrix B, multiply ech element in the i th row of by the corresponding element in the j th column of B, nd then dd these products. The product mtrix B is n m x k mtrix. Mtrix m x n Inner must be equl Mtrix B n x k Outer: Order of B is m x k B = b e f. 2 2 22 2 2 22 b e f e bg _. b e f _ f bh. b e f. ce dg _ b e f. _ cf dh b e f e bg f bh. ce dg cf dh C B b m, n n, k c mk, c k b in nj for i,2,, k nd j,2,, k n
Exmple D-3: Given the mtrices 2 3 b B b 2 22 23 b b b b 2 2 3 3 B b b b 2 22 2 23 3 2 3 Exmple D-4: 3 0 2 B 7 2 3 4 () ( 3)3 (0) ( 3) ( ) ( 3)4 (2) ( 3)( ) 7() 2(3) 7(0) 2() 7( ) 2(4) 7(2) 2( ) 8 3 3 5 3 2 2 Inverse of mtrix: is dj Properties of the Inverse Mtrix: Exmple D-5: I B B 0 0 2 ; 2 2 3 2 0
Properties of Mtrix ddition nd Sclr Multipliction B B Commuttive Property of ddition ( B C) ( B) C ssocitive Property of ddition ( kl) k( l) ssocitive Property of Sclr Multipliction k( B) k kb Distributive Property ( k l) k l Distributive Property 0 0 dditive Identity Property ( ) ( ) 0 dditive Inverse Property Multipliction ( BC) ( B) C ssocitive Property of Multipliction ( B C) B C Distributive Property ( B C) B C Distributive Property