APPENDIX A THE ESSENTIALS OF MATRIX ANALYSIS

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1 APPENDIX A THE ESSENTIALS OF MATRIX ANALYSIS A atrix is a rectangular collection of nubers. If there are η rows and coluns, we write the atrix as A n x, and the nubers of the atrix are α^, where i gives the row nuber and j gives the colun nuber. These are also called the diensions of the atrix. We copactly write A = (α^)\~"'1ΐ. In rectangular for A = an ai 01 «air, O-nl 0, n Gae Theory: An Introduction, First Edition. By E.N. Barron Copyright 008 John Wiley & Sons, Inc. 345

2 346 THE ESSENTIALS OF MATRIX ANALYSIS If η =, we say that the atrix is square. The square atrix in which there are all Is along the diagonal and Os everywhere else is called the identity atrix: In := (f Here are soe facts about algebra with atrices: 1. We add two atrices that have the sae diensions, A + B, by adding the respective coponents A + Β = (a,j + by), or 1 A + B = αιι+&π «1 + &1 α 1 + 6ΐ α + ί> 0-l + h 0 + i» On "Γ" b n O-nl + bnl 0, η + b n. A atrix ay be ultiplied by a scalar c by ultiplying every eleent of A by c; that is, ca = (coy) or ca = can ca i can ca, cai ca n ca n \ ca n 3. We ay ultiply two atrices A n x and B x k only if the nuber of coluns of A is exactly the sae as the nuber of rows of B. You have to be careful because not only is A Β Φ Β A; in general it is not even defined if the rows and coluns don't atch up. So, if A = A n x and Β = B X fc, then C A - Bis defined and C = C n x k, and is given by AB = an 0, We ultiply each row Onl i = 0 1,,. of Β in this way Oln 0n b n 61 bi k 61 6 &fc bl O Ok., n, of A by each colun j = 1,,..., k, iabj = [aii a. bij b j anbij+aibj-\ hai b j = c^-.

3 THE ESSENTIALS OF MATRIX ANALYSIS 347 This gives the (i, jth eleent of the atrix C. The atrix C n *k (cij) has eleents written copactly as Cjj ^ ^ d{ r b r j, i = 1,,..., 7Ί, j 1,,..., k. r=l 4. As special cases of ultiplication that we use throughout this book an a-1 O-ln Xlxn ' A n X T n [xi X Xn] θ1 θ «n «nl 0. n Each eleent of the result is E(X,j), L«=l i=l t=l j = l,,...,. 5. If we have any atrix A n x, the transpose of A is written as A T and is the TO χ η atrix, which is A with the rows and coluns switched: an 0,1 «nl A = «1 «' «n «ln a n p - «n If Yix is a row atrix, then Y T is antoχ 1 colun atrix, so we ay ultiply A x by Y T on the right to get Σ α ΐί% A V T ": ι «1 ai 'yi~ β1 a ", a n i a n On j=l Each eleent of the result is E(i, Y), i = 1,,..., n. 6. A square atrix A n x n has an inverse A 1 if there is a atrix B n x n that satisfies A Β = Β A = I, and then Β is written as A - 1. Finding the inverse

4 348 THE ESSENTIALS OF MATRIX ANALYSIS is coputationally tough, but luckily you can deterine whether there is an inverse by finding the deterinant of A. The linear algebra theore says that A" 1 exists if and only if det(a) Φ 0. The deterinant of a χ atrix is easy to calculate by hand: det (A x; απ a i ai a = «11««101 One way to calculate the deterinant of a larger atrix is expansion by inors which we illustrate for a 3 χ 3 atrix: det(4 3x3; «1 ««3 «11 Ol «13 «31 «3 θ33 an 0,3 «1 a 3 «1 «- ai + ai3 03 «33 a3i «33 «31 a3 This reduces the calculation of the deterinant of a 3 χ 3 atrix to the calculation of the deterinants of three x atrices, which are called the inors of A. They are obtained by crossing out the row and colun of the eleent in the first row (other rows ay also be used). The deterinant of the inor is ultiplied by the eleent and the sign + or alternates starting with a + for the first eleent. Here is the deterinant for a 4 χ 4 reduced to four 3x3 deterinants: det(ajx4) = «11 «1 «13 «14 «1 ««3 «4 θ « «41 «4 «43 «44 = an «θ3 «4 03 «33 «34 j - «1 «4 «43 «44 «1 ««4 +ai3 I «31 «3 «34 «41 «4 «44 01 «3 «4 «31 θ «41 «43 «44 «1 0 «3 ai41 a3i a3 033 «41 «4 «43 7. A syste of linear equations for the unknowns y = (j/i,..., y ) ay be written in atrix for as A nx y = 6, where b = (61,6,, b n ). This is called an inhoogeneous syste if b Φ 0 and a hoogeneous syste if 6 = 0. If A is a square atrix and is invertible, then y = A~ l b is the one and only solution. In particular, if b = 0, then y = 0 is the only solution. 8. A atrix A nxtn has associated with it a nuber called the rank of A, which is the largest square subatrix of A that has a nonzero deterinant. So, if A is a

5 THE ESSENTIALS OF MATRIX ANALYSIS 349 4x4 invertible atrix, then rank(a) = 4. Another way to calculate the rank of A is to row-reduce the atrix to row-reduced echelon for. The nuber of nonzero rows is rank(a). 9. The rank of a atrix is intiately connected to the solution of equations A n x y xi = b n xi. This syste will have a unique solution if and only if rank(a) =. If rank(a) <, then Ay x ι = b nx \ has an infinite nuber of solutions. For the hoogeneous syste Ay x \ = 0 n x i, this has a unique solution, naely, y xi = 0 x i, if and only if rank(a) =. In any other case Ay xi = 0 xi has an infinite nuber of solutions. n