Matrix Algebra. Mathematics Help Sheet. The University of Sydney Business School

Transcription

1 Mtrix Algebr Mthemtics Help Sheet The University of Sydney Business School

2 Introduction Definitions nd Nottion A mtrix is rectngulr rry (or rrngement) of elements tht possesses the generl form: 2 A N 2 22 N 2 K 2K NK A mtrix itself cn be denoted s single cpitl letter such s A, s seen bove. In generl form, ech element within mtrix is denoted s lower cse letter with subscript indicting the row, followed by subscript indicting the column. 2 2 mtrix looks like this: A 3 3 mtrix looks like this: A = [ ] A = [ 4 8] 6 5 The bove two mtrices re clled squre mtrices becuse they possess the sme number of rows nd columns. The Identity Mtrix The identity mtrix is squre mtrix of ny dimension with the vlue forming the elements running digonlly throughout the mtrix from top left to bottom right. It is the mtrix equivlent of the number one such tht ny mtrix multiplied by the identity mtrix results in itself. An identity mtrix possesses the generl form: I N

3 Mtrix Addition nd Subtrction To dd two mtrices, dd ech element with its corresponding element in the other mtrix (or mtrices). Similrly, to subtrct one mtrix from the other, subtrct ech element by the element in the other mtrix in the corresponding position (s subtrcting mtrix is the sme s dding negtive mtrix). [ ] + [2 9 3 ] = [ ] [ ] + [2 5 9 ] = [7 7 6 ] For ddition nd subtrction of mtrices to be possible, the mtrices must hve the sme dimensions. Tht is, they must hve the sme number of rows nd columns. Mtrix Multipliction Sclr Multipliction Sclr multipliction involves multiplying mtrix by constnt. To do so, multiply ech element within the mtrix by the constnt [ 4 2] = [ 2 6 ] Conformbility In order for two mtrices to be multiplied, the number of rows in the first mtrix needs to be equl to the number of columns in the second mtrix. If they stisfy this criteri, then the mtrix product is conformble. In the following exmple, AB is conformble but BA is not, 8 4 A = [ 4 ] 6 B = [ ] 2

4 Mtrix Multipliction To multiply mtrices, you re required to find the dot product of rows nd columns. Finding the dot product involves multiplying ech row of the first mtrix with its corresponding column in the second mtrix nd then summing the multiples up, in order to rrive t single vlue which will form n element in the resulting mtrix. The resulting element s position is determined by the row nd column numbers of its multiples. Tht is, row one multiplied by column one will result in the element in the resulting mtrix. Necessrily then, the dimensions of the resulting mtrix will reflect the number of rows in the first mtrix nd the number of columns in the second. [ ] [ 2] = [ ] 5 To obtin element in the resulting mtrix, the first row of the first mtrix (3 ) is multiplied by the first column of the second mtrix (2 5) such tht =. The sme process is used to find the remining elements, for exmple, element 2 is found by multiplying the first row of the first mtrix nd the second column of the second mtrix. And so on for the remining elements. Conformble Mtrices In order to multiply two mtrices, the number of rows of the first mtrix must equl the number of columns of the second mtrix. Tht is, the mtrices need to be conformble. Thus, the order mtters in mtrix multipliction nd AB is not the sme s BA. Non-conformble Mtrices When ttempting to multiply two mtrices tht re not conformble, the result will be undefined. [ ] [2 5 ] = undefined Alterntive Nottion Alterntive nottion of mtrix multipliction is s follows: n AB ij b jk j 3

5 The bove nottion precisely specifies the multipliction of two mtrices nd my be useful when more compct specifiction is required. Vector Multipliction Vector multipliction is just specil cse of mtrix multipliction where row vector (i.e. mtrix with only row) is multiplied by column vector (i.e. mtrix with only one column). The sme rules of mtrix multipliction s specified bove pply. [5 2 3] + [ 2] = [9] In ll vector multiplictions, the resulting mtrix is single element. Trnspose of Mtrix Trnsposing mtrix involves swpping the rows nd columns such tht the trnspose of mtrix will hve the opposite row nd column dimensions. In the exmple below, 2 3 mtrix becomes 3 2 mtrix. [ ]T + [ 2] 3 A symmetric mtrix is squre mtrix tht is identicl to its trnspose. [ ]T + [ ] A digonl mtrix is symmetric mtrix whose elements off the digonl re ll. 2 3 A = [ ] 3 2 4

6 Finding the Determinnt In order to find the inverse of mtrix, you will need first need to find its discriminnt. The determinnt cn only be found for squre mtrices nd the following sections illustrte how to find the discriminnt of 2 2 nd 3 3 mtrices. The determinnt of mtrix A is generlly denoted s A Discriminnt of 2 2 Given mtrix with two rows nd two columns, A = [ b c d ] The determinnt is clculted by: A = d bc A = [ ] A = A = Discriminnt of 3 3 Given mtrix with three rows nd three columns, b c A = [ d e f] g h i The determinnt is clculted by: A = (ei fh) b(di fg) + c(dh eg) 2 A = [ 3 2] A = 2( ) ( 4 2 3) + ( 6 3 3) A = 2(2) + (9) A = 3 5

7 Finding the Inverse The inverse of mtrix is conceptully the sme s the reciprocl of number, such tht if you multiply mtrix by its inverse, the result will be n identity mtrix. If mtrix hs non-zero determinnt, then tht mtrix will hve unique inverse. Hence, while only squre mtrices cn hve inverses, not ll squre mtrices hve n inverse. The inverse of mtrix A is generlly denoted s A To clculte the inverse of 2 2 mtrix: A c b d A d c b A = [ ] Step : Find the determinnt: A = A = Step 2: Find the inverse A = [ ] = [ ] A = [ ] To check tht you hve found the right nswer, you cn multiply the inverse with the originl mtrix. If the resulting mtrix is n identity mtrix, then you hve obtined the correct inverse. To check the exmple bove, AA = [ ] [ ] AA = [ ] 6