Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

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1 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the numers re clled entries of the mtrix. Mtrices re usully denoted y uppercse letters:,, nd so on. Hence, 5 6 re mtrices. lerly, mtrices come in vrious shpe depending on the numer of rows nd columns. For exmple, the mtrix shown hs rows nd columns. In generl, mtrix with m rows nd n columns is referred to s n m n mtrix or s hving size m n. hus mtrices,, ove hve sizes,,, respectively. mtrix of size n is clled row mtrix, wheres one of size m is clled column mtrix. Ech entry of mtrix is identified y the row nd column in which it lies. he rows re numered from the top down, nd the columns re numered from left to right. hen the ( i, j) - entry of mtrix is the numer lying simultneously in row i nd column j. For exmple: he (, ) entry of is 5 6 Liner lger I 4

2 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition he (, ) -entry of is specil nottion hs een devised for the entries of mtrix. If is n m n mtrix, nd if the ( i, j) - entry is denoted s ij s follows:, then is displyed m m n n mn his is usully denoted simplify s ij. n n n is clled squre mtrix. For squre mtrix, the entries : the min digonl of the mtrix.,, nn, re sid to lie on wo mtrices nd re clled equl ( written = ) if nd only if :. hey hve the sme size. orresponding entries re equl or cn e written s ij ij mens tht ij ij for ll i, j. Exmple Given, c d, 5 6, discuss the possiility tht,, : = is impossile, ecuse nd re of different sizes. Similrly, is impossile. is possile provided tht corresponding entries re equl: c d = mens,, c, d. Liner lger I 5

3 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix ddition If nd re mtrices of the sme size, their sum formed y dding corresponding entries. If ij nd ij is the mtrix, this tke the form: ij ij Note tht ddition is not defined for mtrices of different sizes. Exmple If nd 4, compute! 5 6 = Exmple Find,, c if c c dd the mtrices on the left side to otin: c c ecuse the corresponding entries must e equl, this gives three equtions: c,, c. Solving these yields,, c. he properties of Mtrix ddition If,, re ny mtrices of the sme size, then:. ( commuttive lw ). ( ) ( ) ( ssocitive lw ) he m n mtrix in which every entry is zero is clled the zero mtrix nd is denoted s, hence, Liner lger I 6

4 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition. X X he negtive of n m n mtrix ( written s - ) is defined to e m n mtrix otined y multiply ech entry of y. If ij, this ecomes ij, hence, 4. ( ) for ll mtrices ij where is the zero mtrix of the sme size s. closely relted notion is tht of sutrcting mtrices. If mtrices, their difference ( ), i.e. : is defined y: ij ij ij ij, re two m n Exmple 4,, ompute,, = 5 Exmple 5 Solve + X =, where X is mtrix. X must e mtrix. If X = x u y v, the eqution reds: Liner lger I 7

5 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition = + x u y v = x u y v he rule of mtrix equlity gives x, y, u, v. hus X =. We solve numericl eqution x y sutrcting the numer from oth sides to otin x. his lso works for mtrices. o solve + X =, simply sutrct the mtrix from oth sides to get: X = - Sclr Multipliction In Gussin Elimintion, multiplying row of mtrix y numer k mens multiplying every entry of tht row y k. More generlly, if is ny mtrix nd k is ny numer, the sclr multiple k is the mtrix otined from y multiplying ech entry of y. If ij,this is: k k ij We hve een using rel numers s sclrs, ut we could eqully well hve een using complex numers. Exmple 6 If,nd, compute 5, Liner lger I 8

6 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition 5 = ( ) = If is ny mtrix, note tht k is the sme size s for ll sclrs k. We lso hve: nd k ecuse the zero mtrix hs every entry zero. In other words, k if either k or. he properties of sclr multipliction Let, denote ritrry m n mtrices, where m, n re fixed, let k, l re denote ritrry rel numers. hen :. k ( ) k k. ( k l) k l. ( kl ) k( l) 4.. Exmple 7 Simplify ( ) ( ) [ ( 4) 4( )] where,, re ll mtrices of the sme size. ( ) ( ) [ ( 4) 4( )] rnspose Mny result out mtrix involve the rows of, nd the corresponding result for columns is derived in n nlogous wy, essentilly y replcing Liner lger I 9

7 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition the word row y the word column throughout. he following definition is mde with such ppliction in mind. If is n m n mtrix, the trnspose of, written s mtrix whose rows re just the columns of in the sme order. In other words, the first row of of is the second column of, nd so on., is the is the first column of, the second row n m Exmple 8 Write down the trnspose of ech of the following mtrices:,,, D 4,,, D 4 If ij is mtrix, write ij. hen ij is the j th element of the i th row of nd so is the j th element of the i th column of. his mens ij ji so the definition of cn e stted s follows: If ij, then ji he Properties of trnsposition: heorem Let, denote mtrices of the sme size, nd let k denote sclr.. If is n m n mtrix, then ). ( is n n m mtrix.. ( k) k Liner lger I

8 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition 4. ( ) he mtrix D in Exmple 8 hs the property tht re importnt. mtrix is clled symmetric if D D.. Such mtrices symmetric mtrix is necessrily squre. he nme comes from the fct tht these mtrices exhiit symmetry out the min digonl. ht is, entries tht re directly cross the min digonl from ech other re equl. Exmple 9 If, re symmetrics n n, show tht is symmetric We hve ( ) nd, so y heorem, we otin Exercises.:. Find,, c, d if:. c d c d d d.. Let,, 4, D, 4 E. ompute the following ( if possile).. D E c. ( ) d. 4D e. D E. If X, Y, nd re mtrices of the sme size, solve the following equtions to otin X, Y in terms of, :. 5X X Y Y 4X. 5X Y 4Y Liner lger I

9 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition 4. squre mtrix is clled skew-symmetric if. Show tht is skew-symmetric. Let e ny squre mtrix.. Find symmetric mtrix S nd skew-symmetric mtrix W such tht S W c. If W is skew-symmetric show tht the entries on the min digonl re zero 5. squre mtrix is clled digonl mtrix if ll entries off the min digonl re zero. If, re digonl mtrices show tht,, k re digonl mtrices. 6. Let e ny squre mtrix. If p nd q for some mtrix nd numers p, q, show tht either or p. q Liner lger I