] 3 ] 3 0. Exercises with Matrices. Part One Practice with Numbers (if there is no answer, say so) Solve for x and y:

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1 Keeth L. Simos, 005 Eercises with Matrices Part Oe Practice with Numbers (if there is o aswer, say so) [ ] [ ] [ 7 ] [ 4 7 ] Note: The first matri above is called a permutatio matri. See how it permutes the rows of the right-had matri? A permutatio matri is a square matri that cosists of all 0s or s, with a sigle i each row ad a sigle i each colum T T T Solve for ad y: = y y 6. Solve for ad y: 3 4 = Questios about determiats. I each case, fid the determiat ad idicate whether the matri is sigular. Note that, for eample, = a. b. c. d Part Two Types of Matrices Give the most specific ame for each matri, from the followig choices: colum

2 Keeth L. Simos, 005 matri, diagoal matri, idetity matri, lower triagular matri, permutatio matri, row matri, square matri, symmetric matri, upper triagular matri. a b [ ] a b c Part Three Uderstadig Symbolic Matri Maipulatio I this part, thik of, y, ad X as matrices cotaiig ecoomic data. The sample mea or average of umbers is =. The sample variace of the umbers is a i = i measure of how much they vary: σ = (i ). The sample covariace i = compares two differet kids of umbers, such as icome ad health, ad is a measure of how much they vary together: σ y = (i )(yi y). You should recogize i = these cocepts i some of the algebraic epressios you study below. k Let,, i = k = y = [ y y y ], ad X =. k For each item below, write out how the aswer (usually a matri) looks i detail, usig ellipses ( ) where ecessary. What is: 9. ii 39. ( M) ( M) 33. I ii i y ii 34. M I ii 4. ( M) ( My) 4. ( M ) ( My) 3. ii 35. i 36. i ( M ) 43. MX 3. I ii 37. MM 44. ( MX) ( MX) 38.

3 Keeth L. Simos, ( MX ) ( MX ) Part Four Symbolic Matri Maipulatio Simplify all epressios. Assume that matrices A ad B are symmetric. Assume that iverses eist for matrices A, B, C, ad D. (By the way, ote that the iverse of a symmetric matri is symmetric.) 46. I 47. C C 48. ( C ) XX XX) 49. ( ) 50. ( CD) 5. ( CD) ( 5. A A 53. C C 54. B ( AB) A Part Five Quadratic Forms For each of the followig epressios, determie whether it is a quadratic form. If it is, write the epressio i the matri form A, where is a vector ad A is a symmetric matri. Greek letters deote parameters that do ot ivolve the s π 63. δ α +β +γ 3 + +φ 3 +σ ρ 3 Part Si Matri Derivatives If you have forgotte what derivatives are or basic formulae for derivatives, please review them. Total versus partial derivatives: I a total derivative, you take ito accout that oe variable may be a fuctio of aother; for eample, if c = b ad dƒ(b,c) ƒ(b, c) = b + 3c, the = b + 6. I a partial derivative, you deliberately hold db costat all variables but oe; for eample, ƒ(b,c) = b because you deliberately b

4 Keeth L. Simos, assume that c is beig held fied (by meas which deped o the cotet). Key rules to master: Kow by memory basic rules for derivatives, icludig evetually the chai rule. ku 7 07 (a + bu ) l(ku) (u e ) (u ) ep(k(l u) ) Self test:,,,, (aswers o last page). The fial oe of these epressios defiitely requires the chai rule. As log as you kow these basics, matri derivatives are ot too hard, especially u = u u, ad ƒ( u) is a fuctio what we do here. Suppose u is a vector, [ ] whose value depeds o u,,u. If you eed to kow all derivatives ƒ( u ) ƒ( u ),,, it is coveiet to write all the derivatives i a sigle vector. The ƒ( u) ƒ( u) vector is writte, ad is equal to. The et two questios show you two ƒ( u) useful formulae (used i chapter of Hayashi) for computig such vector derivatives. a a a, ad let u = [ u u ]. First, write out au i terms of the elemets a through a ad u through u (your au aswer should be a scalar). Secod, compute the derivative. Third, write the 64. Let a be a vector of costats = [ ] au au whole vector of derivatives. You should have just proved that ( au ) = a. 65. Let A = a a, ad let u a a u = u. Prove that ( uau ) = Au if A is symmetric, ad prove that this formula does ot work if A is ot symmetric. (This formula i fact works for ay symmetric matri A ad vector u.) Above you eeded to differetiate a sigle value ( scalar ), which we called ƒ( u), with respect to each elemet i a vector, which we called u, givig derivatives arraged i a vector. The same idea eteds to differetiatig m differet values, i a vector f, with respect to each elemet i the vector, u, givig m ƒ( u) derivatives arraged i a matri. Let f = be a vector cotaiig the m differet ƒ m ( ) u values, which may be formulae ivolvig u ad hece have bee writte f ƒ( u),,ƒ m( u). The epressio meas the followig matri of derivatives: u i

5 Keeth L. Simos, ƒ( u) ƒ( u). This matri has m rows ad colums, i.e. oe row for each ƒ m( u) ƒ m( u) etry i f ad oe colum for each variable i u. Note that u is trasposed at the bottom of the epressio f, thus emphasizig that the values u,,u differ from left to right f (ot top to bottom) i the matri. (Some authors write to mea the same matri, but Hayashi s tetbook uses f sice it is clearer i this regard.) u + 4u + 6u3 u 66. Let f = 3 4, ad let 00 + uuu3 u = u. Compute the matri f, writig out u 3 each of the derivatives i the matri. Fially, if you start with a sigle value ( scalar ) ƒ( u) ad differetiate twice with respect to each elemet i u, it is coveiet to arrage the resultig derivatives i a ƒ( u) ƒ( u) ƒ( u) u u matri. The epressio meas the matri u. This matri ƒ( u) ƒ( u) is give the ame Hessia matri, which you will see sometimes. This matri is symmetric as log as the secod derivatives i it eist ad are cotiuous (at the values of ƒ( ) ƒ( ) u beig cosidered), because the Clairaut s theorem proves that u = u. The Hessia matri is used i maimizatio ad miimizatio. i j j i u 4 ƒ( u) 67. Let ƒ( u) = u + uu3, where u = u. Compute the Hessia matri u u, 3 writig out each of the derivatives i the matri. v 68. Let v = v, ad let A be the matri you computed i the previous questio. Write v 3 out vav. Maimizatio ad miimizatio problems ca be solved (if first ad secod derivatives eist) usig first ad secod order coditios. Suppose you are maimizig

6 Keeth L. Simos, or miimizig ƒ(u), by choosig the values of u,,u. The first order coditio is that ƒ( u) ƒ( u) each of the derivatives i must be zero, i.e., = 0. (Sice a derivative is a slope, this just says that the slope of the fuctio must be completely flat, i each of the directios, at the chose values of u,,u, just as there is a flat spot at the top of Mout Everest or at the bottom of the Mariaa Trech). Solvig these equatios gives values for u,,u. The secod order coditios, always applied after solvig the first order coditio, comes i differet versios. Versio A. If met, this coditio guaratees you have foud a global * maimum (miimum). The coditio is that f(u) is cocave (cove) everywhere, i.e. that the fuctio is dome-shaped or bowl-shaped. This is ƒ( u) guarateed if for all possible values of u, the Hessia matri is egative u (positive) semidefiite. For ay matri A to be egative (positive) semidefiite meas that vav is always 0 ( 0 ) for ay possible vector of real umbers v. You do t eed to kow how to determie this, but if you are iterested it is discussed i tetbooks o quatitative methods for ecoomics or i liear algebra tets. To ƒ( u) summarize, to check versio A of the secod order coditio, compute A = u ad the see if vav 0 ( 0 ) whe you plug i all possible values of u ad v. Versio B. If met, this coditio guaratees you have foud a local maimum (miimum), which might be the global maimum or miimum (oe should also check what happes as the values i u become egative or approach ay boudaries or discotiuities). The coditio is that f(u) is cocave (cove) immediately aroud the poit you foud usig the first order coditio. This is guarateed if, for the specific ƒ( u) poit u you foud, the Hessia matri is egative (positive) defiite. For ay u matri A to be egative (positive) defiite meas that vav is always <0 (>0) for ay possible vector of real umbers v, with the eceptio of a vector of zeros v=0. Agai, you do t eed to kow how to determie this. To summarize, to check versio B ƒ( u) of the secod order coditio, compute A =, plug i = u u u i your aswer, ad the see if vav < 0 (>0) whe you plug i all possible values of v. If you are checkig versio B ad vav is sometimes <0 ad other times >0 depedig o the values i v, the matri A is called idefiite ad the solutio foud is either a maimum or a miimum (it might be a saddle poit i the fuctio ƒ(u)). There are further ways to check the secod order coditio, but these are more advaced topics. * Global maimum meas the highest maimum aywhere; global miimum is the lowest miimum aywhere. You could alteratively have a local maimum or miimum; for eample the peak of Mout Washigto is a local high poit but ot the highest poit i the world.

7 Keeth L. Simos, I the previous questio you wrote out vav, where A was a Hessia matri. At the u 0 poit where u = 7, is vav >0, <0, 0, or 0 for all possible values of v 0? u Suppose you wat to choose values of u that miimize the fuctio ƒ( u) = u + uu3. Write the first order coditio ad solve for the values of u that satisfy this coditio. You have already computed the Hessia matri for the fuctio, so as you did i the previous questio, see whether vav satisfies versio B of the secod order coditio. I.e., usig versio B of the secod order coditio, ca you cofirm that you have foud a local miimum? Aswers to Self-Test for Derivatives: (a + bu ) = bu l(ku) = u ku (u e ) ku ku ku = (u)e + u (ke ) = (ku + u)e 7 6 (u ) (7u ) 5 = = 4u 07 ep(k(lu) ) = ( ep(k(lu) ))( 07k(lu) ) u If you had trouble oly with the last questio, the sometime over the first moth of the semester it is a good idea to use my derivatives eercises eough to get used to the chai rule. If you had trouble with multiple questios, it is a good idea to practice just the very simple derivative rules right away.