(IC) Matrix algebra Before formalizig Gauss-Jorda i terms of a fixed procedure for row-reducig A, we briefly review some properties of matrix multiplicatio Let m{ [A ij ], { [B jk ] p, p{ [C kl ] q be matrices, with etries i (say) R, or more geerally ay field (cf IIA) Recall that the traspose t A is the m matrix with etries ( t A) ij = A ji I write the superscript t o the left so that later we ca talk about the traspose iverse t A without paretheses The m m idetity matrix with etries if i = j ad 0 otherwise will be deoted by I m (or just I); ad we will write E m ij (or just E ij) for the matrix with (i, j) th etry ad all other etries zero Multiplicatio: At ay rate, we defie the matrix product AB to be the m p matrix with etries (AB) ik := A ij B jk j= Associativity of this product follows from associativity of the groud field: ((AB)C) il := k = j,k (AB) ik C kl = k ( j A ij (B jk C kl ) = = (A(BC)) il A ij B jk )C kl = (A ij B jk )C kl j,k
2 (IC) MATRIX ALGEBRA Commutativity fails: BA is ot eve defied uless p = m, i which case the closest oe has is BA = t ( t A t B) A example where A ad B are actually symmetric: [ ] [ ] [ ] [ ] [ ] [ ] 0 0 0 0 0 0 0 = = = 0 0 0 0 0 0 0 0 0 For a physicist, ocommutativity is essetial, sice it s the etire poit of the Heiseberg ucertaity priciple that the positio ad mometum operators do t commute! Or as the Mad Hatter says, seeig what you eat ad eatig what you see are ot at all the same thig Iverses: if A is m for m <, it caot have a left iverse (L such that LA = I ) but may have may right iverses (R such that AR = I m ) A example, where a, b ca be ay real umbers: A {[ }} ]{ t 0 0 0 at bt a b 0 = [ 0 0 If m > the the situatio is just reversed For square matrices (m = ) we will prove i ID that { of a left iverse} { of a right iverse} But if both exist for a matrix A, they must be the same: BA = I, AC = I = B = BI = B(AC) = (BA)C = IC = C I this situatio we say A is ivertible, deotig the (left ad right) iverse matrix by A Products ad iverses of ivertible matrices are ivertible; eg for products AB, B A furishes a 2 -sided iverse EXAMPLE If αδ βγ = 0 (for α, β, γ, δ i your favorite field), ] we have ( α γ β δ ) = ( αδ βγ δ γ β α ) though puttig these words i the Mad Hatter s mouth may have bee a polemic o Lewis Carroll s part agaist the quaterios
(IC) MATRIX ALGEBRA 3 Vectors ad matrix multiplicatio: For x R, here are some characterizatios of the matrix-vector product A x i terms of rows ad colums of A : = c A x = c c c x x x = r x r m x = x c + + x c = x i c i i= Writig ê i for the coordiate vectors of R, we see that Aê i = c i ad so A = Aê Aê (With this uderstood, you should ow be able to easily covice yourself that the colums of a matrix product AB are liear combiatios of the colums of A!) There are two differet ways to multiply vectors as matrices: 3 [ 2 ] = 4 = dot (iterior) product, 2 6 2 2 [ 3 ] = 3 = exterior product 3 I particular, if x ad y are two colum vectors, the the dot product x y i terms of matrix multiplicatio is t X Y (where X ad Y are the correspodig m matrices)
4 (IC) MATRIX ALGEBRA Elemetary matrices: These are m m (square) matrices of oe of the followig three types: Sij m := I m Eii m Em jj + Em ij + Em ji = 0 0, Si m ( a ) := I m + ( a )E ii = a, ad R m ji ( b) := I m be m ji = b I ll drop the superscript m i the sequel sometimes The elemetary row operatios of IB may be iterpreted as left-multiplyig the augmeted matrix (represetig our liear system) by oe of these elemetary matrices: the first exchages the i th ad j th rows of the matrix it operates o; the secod divides the i th row by a; ad the third subtracts b (i th row) from the j th row All three types of matrices are clearly ivertible, with S ij = S ij, S i ( a ) = S i (a), ad R ji ( b) = R ji (b) Exercises () Fid two differet 2 2 matrices A such that A 2 = 0 but A = 0 (2) By carryig out Gauss-Jorda ad keepig track of your steps, fid elemetary matrices E,, E k such that E k E 2 E A = I,
where A := EXERCISES 5 2 0 3 0 (3) Let A be a upper triagular m m matrix (That is, A ij = 0 for i > j) Show that A is ivertible if ad oly if all the diagoal etries A ii are ozero (4) Cosider the set H M 2 (C) (of 2 2 matrices with complex etries) of the form ( ) ( α β a x = = 0 + a β ᾱ a 2 + a 3 a2 + a 3 a0 a ), a i R Show that H is closed uder additio ad multiplicatio, ad that every ozero elemet is ivertible show that multiplicatio is ot commutative Give a example to