Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous work tht Ax = b hs unique solution for ny n right side b if nd only if rref(a) = I n, the n n identity mtrix It follows tht if rref(a) = I n, then rref([a b]) = [ I n c] nd x = c is the unique solution of Ax = b In the cse where b = 0, tht is, homogeneous liner system, x = c = 0 which implies the liner system hs only the trivil solution Coefficient mtrices of squre liner systems with unique solutions re importnt in their own right nd we mke the following definition Definition An n n mtrix A is clled nonsingulr or invertible provided rref(a) = I n Thus we sy tht nonsingulr liner system hs unique solution, mening tht the coefficient mtrix of the liner system is squre nd its RREF is n identity mtrix Next we develop n importnt property of nonsingulr mtrices We will use liner combintions, liner systems, RREF, nd properties of prtitioned mtrices Recll tht mtrix product PQ cn be viewed s P times Q prtitioned into its columns, tht is, Q = [q q 2 q n ], so PQ = P[q q 2 q n ] = [Pq Pq 2 Pq n ] Thus ech column of the product is relly liner combintion of the columns of mtrix P Hill\Kolmn Fll 999 Section 24 Pge
If A is n n n nonsingulr mtrix, then for ny n mtrix b, the liner system Ax = b hs unique solution Prtition I n into its columns which we denote s e j, j =, 2,, n; I n = [e e 2 e n ] Then Ax = e j hs unique solution for ech j Denote the solution s c j Hence we hve () Ac = e, Ac 2 = e 2,, Ac n = e n Writing () in mtrix form, we hve [Ac Ac 2 Ac n ] = [e e 2 e n ] = I n From our observtion on prtitioned mtrices bove, this eqution cn be written in the form A[c c 2 c n ] = I n Letting C = [c c 2 c n ], we hve just shown the following result Sttement I If A is nonsingulr then there exists unique mtrix C such tht AC = I n Note tht vector c j = col j (C) is the unique solution of Ax = e j, hence it follows tht mtrix C is unique It is lso true tht Sttement II If A is nonsingulr then there exists unique mtrix D such tht DA = I n Next we show tht C = D s follows: (2) C = I n C = (DA)C = D(CA) = DI n = D Thus combining Sttements I nd II we hve Sttement III If A is nonsingulr then there exists unique mtrix C such tht AC = CA = I n Hill\Kolmn Fll 999 Section 24 Pge 2
Sttement III motivtes the following definition Definition For n n n nonsingulr mtrix A, the unique mtrix C such tht AC = CA = I n is clled the inverse of A We denote the inverse of A by the symbol A - Now you see why nonsingulr mtrices re lso clled invertible mtrices; they hve n inverse Wrning: A - is not to be interpreted s reciprocl ; it is just the nottion to denote the inverse of mtrix A To complete this chrcteriztion of nonsingulr mtrices we lso need to verify tht if A hs n inverse then rref(a) = I n We do this t the end of this section To summrize things we present the following set of equivlent sttements: A is nonsingulr 2 rref(a) = I n 3 A hs n inverse We hve tied together the unique solution of squre liner system, RREF, nd the mtrix multipliction property AA - = A - A = I n This leds us to the following observtion A nonsingulr liner system Ax = b hs the unique solution x = A - b This follows from the equivlent sttements given next: Ax = b or A - A x = A - b or I n x = A - b or x = A - b (3) The reltions in (3) sy tht we hve formul for the unique solution of nonsingulr liner system nd it cn be computed in two steps: i) Find A - ii) Compute the mtrix-vector product A - x In the rel or complex numbers, the term reciprocl mens inverse Hill\Kolmn Fll 999 Section 24 Pge 3
The finl result in (3) is dvntgeous for lgebric mnipultion of expressions nd theoreticl purposes, but the solution of Ax = b is more efficiently computed using the fct tht rref([a b]) = [ I n c] implies tht the unique solution is x = c Next we show how to compute the inverse of mtrix The procedure we develop simultneously determines if squre mtrix hs n inverse, nd if it does, computes it From () we hve tht the columns of the inverse of A re the solutions of the liner systems Ax = col j (A) = e j, j=,2,, n The corresponding set of ugmented mtrices cn be combined into the following prtitioned mtrix [A e e 2 e n ] = [A I n ] From our previous rguments we hve the following sttement: Sttement IV For n n mtrix, rref([a I n ]) = [ I n C] if nd only if A is nonsingulr nd C = A - It follows tht if rref([a I n ]) [ I n C], then A hs no inverse We cll mtrix with no inverse singulr mtrix Recll tht our focus here is on squre mtrices The terms singulr nd nonsingulr do not pply to mtrices which re not squre Exmple 2 For ech of the following mtrices determine if it is nonsingulr, nd if it is, find its inverse ) A = 4 5 2 9 0 3 3 6 Hill\Kolmn Fll 999 Section 24 Pge 4
2 b) W = 3 4 8 3 Exmple 3 Let A = c b d be nonsingulr By crefully using row opertions on [A I I 2 ] we cn develop n expression for A - s follows To strt, we know thtrref(a) = I 2 ; hence one of the entries in column of A is not zero Assume tht 0; if = 0, then we cn interchnge rows nd 2 below nd proceed Use the entry s the first pivot nd perform the following sequence of row opertions: b 0 c d 0 Since rref(a) = I 2 entry d b 0 c d 0 R cr R2 + bc d bc = 0 b 0 d- bc 0 - c (Why must this be true?) so we cn use it s our second pivot to continue the reduction to RREF b 0 d- bc 0 -c d bc R 2 b 0 -c d- bc 0 d- bc Next we eliminte bove the leding in the second row: Thus b 0 -c d - bc d A = d bc c d bc 0 d - bc b R 2 R + b d bc = d bc d bc 0 0 d c d d - bc -c d - bc b -b d - bc d - bc Hill\Kolmn Fll 999 Section 24 Pge 5
To use this formul we note tht we compute the reciprocl of (d - bc), interchnge the digonl entries, nd chnge the sign of the offdigonl entries Following this prescription for A = 3 2 0 = 0 0 (3)(0) (2)( ) 2 3 = 2 2 3 3 2 0 we hve We lso note tht if d - bc = 0, then the 2 2 mtrix hs no inverse; tht is, it is singulr This computtion, d - bc, is often used s quick method to test if 2 2 mtrix is nonsingulr or not We will see this in more detil in Chpter 3 For 3 3 mtrices development similr to tht in Exmple 3 cn be mde The result is good del more complicted Let w =ie-fh-idb+dch+gbf-gce Then b c d e f g h i = w ie fh (ib ch) bf ce id + fg i cg (f cd) ( dh + eg) (h bg) e bd) If the vlue of w = 0, then the 3 3 mtrix is singulr Computtionlly it is more relistic to compute rref([a I ]) nd extrct A - s we did in Exmple 2 Hill\Kolmn Fll 999 Section 24 Pge 6
The computtion of the inverse of nonsingulr mtrix cn be viewed s n opertion on the mtrix nd need not be linked to the solution of liner system The inverse opertion hs lgebric properties in the sme wy tht the trnspose of mtrix hd properties We dopt the convention tht if we write the symbol A -, then it is understood tht A is nonsingulr Using this convention we next list properties of the inverse Algebric Properties of the Inverse i) (AB) - = B - A - The inverse of product is the product of the inverses in reverse order ii) (A T ) - = (A - ) T When A is nonsingulr so is A T nd the inverse of A T is the trnspose of A - iii) (A - ) - = A When A is nonsingulr so is A - nd its inverse is A iv) (ka) - = (/k) A - k 0 When A is nonsingulr, so is ny nonzero sclr multiple of it v) (A A 2 A r ) - = A r - A 2 - A - The inverse of product extends to ny number of fctors The verifiction of the preceding properties uses the fct tht mtrix C is the inverse of n n n mtrix D provided CD = I n, or DC = I n This is the lterntive to the definition of nonsingulr mtrices given in Sttement III erlier We illustrte the use of this equivlent chrcteriztion of nonsingulr mtrices in Exmple 4 where we verify i) nd leve the verifiction of the remining properties s exercises Exmple 4 Show tht (AB) - = B - A - Hill\Kolmn Fll 999 Section 24 Pge 7
A common thred tht hs linked our topics in this chpter is liner combintions It is unfortunte tht the opertion of computing the inverse does not follow one of the principles of liner combintions Nmely (A + B) - is not gurnteed to be A - + B - We illustrte this filure nd more in Exmple 5 Exmple 5 ) Let A = nonsingulr (verify), but A 0 0 0 0 b) Let A = 0 0 nd B = 2 0 nd B = 6 0 0 8 + B= 0 2 I 2 So A + B is singulr 0 0 0 However, A + B = I 2 is nonsingulr (Verify) Both A nd B re is not since its RREF is Both A nd B re singulr (Verify) Hence nonsingulr mtrices re not closed under ddition, nor re singulr mtrices As we hve seen, the construction of the inverse of mtrix cn require quite few rithmetic steps which re crried out using row opertions However, there re severl types of mtrices for which it is esy to determine by observtion if they re nonsingulr We present severl such results next Sttement V A digonl mtrix, lower tringulr mtrix, nd n upper tringulr mtrix re nonsingulr if nd only if ll the digonl entries re different from zero Hill\Kolmn Fll 999 Section 24 Pge 8
For nonsingulr digonl mtrix D we cn explicitly construct its inverse, D -, s we show in Exmple 6 Exmple 6 Let D be n n n nonsingulr digonl mtrix By Sttement V we hve D = d 0 0 0 0 d22 0 0 0 0 % % 0 0 0 0 d nn where d jj 0, j =, 2,, n To construct D - we compute rref([d I n ]) Since the only nonzero entries re the digonl entries we use row opertions to produce leding 's It follows tht the n row opertions 2 d d22 dnn R, R, R rref([d I ]) = [I D ] = n, pplied to [D I n ] give n n d 0 0 0 0 0 0 d22 0 0 0 0 0 0 0 0 % 0 0 % % 0 % 0 0 0 0 0 0 0 Hence D - is digonl mtrix whose digonl entries re the reciprocls of the digonl entries of D dnn Hill\Kolmn Fll 999 Section 24 Pge 9
We next investigte the subspces ssocited with nonsingulr mtrix The key to verifying ech of the following sttements is tht the RREF of nonsingulr mtrix is n identity mtrix Let A be n n n nonsingulr mtrix, then we hve the following properties Properties of Nonsingulr Mtrix A i) A bsis for row(a) ( col(a) ) re the rows ( columns ) of I n ii) ns(a) = 0 iii) The rows ( columns ) of A re linerly independent iv) dim(row(a))=n nd dim(col(a)) = n v) rnk(a) = n It is often difficult to determine by inspection whether squre mtrix A is singulr or nonsingulr For us, t this time, the determintion depends upon looking t the RREF However, since the RREF of nonsingulr mtrix must be n identity mtrix sometimes n inspection of A cn tell us tht this is not possible, hence we cn clim tht A is singulr The following is list of occurrences tht immeditely imply tht A is singulr Wys to Recognize Singulr Mtrices i) A hs zero row or zero column ii) A hs pir of rows (columns) tht re multiples of one nother iii) As we row reduce A, zero row ppers iv) The rows ( columns ) of A re linerly dependent v) ns(a) contins nonzero vector Hill\Kolmn Fll 999 Section 24 Pge 0