PROBLEM SET I (Suggested Solutions)

Transcription

1 Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x 3 = 6 = Recall that, for ay square matrix = a, if i=, j= ( ) ( ) after we delete the ith row ad jth colum of, the scalar det ( ) the sub-matrix that is left = is defied as the (i,j)th mior of. The x sub-matrix of that is obtaied whe the first rows ad colums of are retaied is called the th-order leadig pricipal mior of : = : =,,..., Let π = ( π, π,.., π ) a permutatio of the itegers { } permutatios of the itegers {,,..., }. Deote by π the symmetric x matrix,,..., ad Π the set of all obtaied by applyig the permutatio π to both the rows ad the colums of : aππ a ππ. a ππ π = aππ a. a ππ ππ

2 Eco3-Fall3 For {,,.., }, let π deote the x submatrix of π obtaied by retaiig oly the first rows ad colums: aππ a ππ. a ππ π = aπ a. a π ππ ππ For {,,.., } ad π Π, π is a th-order pricipal mior of. real, symmetric matrix leadig pricipal miors are positive: > : =,,..., = a is positive defiite if ad oly if all of its i=, j= O the other had, a real symmetric matrix = a is positive semi-defiite if i=, j= ad oly if π for all =,,..., ad for all π Π. real symmetric matrix = a is egative defiite if ad oly if the sequece i=, j= of its leadig pricipal miors alterate i sig startig with the first oe beig egative (or, equivaletly, - is positive defiite): ( ) > : =,,..., O the other had, a real symmetric matrix ad oly if ( ) = a is egative semi-defiite if i=, j= π for all =,,..., ad for all π Π. a) Cosider the leadig pricipal miors of the give matrix: = 3< 3 4 = = 8 6 = > 4 6 The give matrix is egative defiite.

3 Eco3-Fall3 b) = > = = = > The give matrix is positive defiite. c) = > = = 4 4= = 4 5 = x x + = ( 4 5) x( ) = 5 < The give matrix is either positive or egative (semi-)defiite. Its defiiteess caot be determied (idefiite). d) = > = = > 3 3 = = x + + 3x = ( 8 ) + 3x( 6) = 8 8 = < ote that we eed ot proceed to calculate the fourth pricipal mior. Sice oe of the pricipal miors came out egative, the matrix caot be positive defiite. Hece, the give matrix is idefiite. 3. x matrix = a will have as may th-order pricipal miors as there are i=, j=! permutatios π of itegers out of the total of. Hece, it will have th-!! order pricipal miors for =,,...,. ( ) 3

4 Eco3-Fall3 4. x x Cosider some vector x R : x =.. x T The quadratic form Q = x x ca be writte aalytically as follows: x x T Q = x x = ( x x x ). a a. a x ml m l ad some x square matrix [ a ] =, = =. ( a x a x.. a x a x a x.. a x a x a x.. a x ) = ( ax axx.. axx) ( axx ax.. axx ) ( axx axx ax) = Let us evaluate ow this quadratic form alog the coordiates of R. Cosider the vector correspodig to the dimesio alog the first coordiate x =... We get T Q = x x = ( ) = ( a, a, a ) = a.. a a. a Similarly for the vector correspodig to the dimesio alog the mth coordiate m. x = (this vector has a etry of at the mth row ad zeros everywhere else). x x. x 4

5 Eco3-Fall3 We get mt m.. Q = x x = (.. ) = ( a, a, a ) = a a a. a.. m m m mm We are give that the quadratic form Q is positive defiite. Hece, we ought to have m Q > for ay x R. Thus, it ought to be Q > also for ay x R : m=,,.., This gives amm > : m=,,..,. Hece, we establish that Q > amm > : m=,,..., i.e. a ecessary coditio for the geeric square matrix to be positive defiite is that all its diagoal etries a mm to be positive. I the case where the quadratic form Q is positive semi-defiite, we have Q for ay m x R. Thus, it ought to be Q also for ay x R : m=,,.., This gives amm : m=,,..,. Hece, we establish that Q amm : m=,,..., i.e. a ecessary coditio for the geeric square matrix to be positive semi-defiite is that all its diagoal etries a mm to be o-egative. Oe ca follow the above proof for the case where the matrix is egative (semi-) defiite. It is easy to establish that: Q< amm < : m=,,..., ad Q amm : m=,,...,. Hece, a ecessary coditio is that all the diagoal etries are egative (o-positive) respectively. To show that the above coditios are ot sufficiet, cosider the matrix i part (c) of problem above. For this matrix, all of its diagoal etries are positive, yet it is ot positive defiite. 5

6 Eco3-Fall3 5. m Let x, x R ad λ R. From the defiitio of h: R h λx + λ x = (( ( ) )) ( λ ( λ) ) ( + + ) = ( λ( ) ( λ)( )) f x x b f x + b + x + b ( + ) + ( λ) ( + ) = ( ) + ( λ) h( x ) ( ) λf x b f x b λh x Hece, we establish that ( ) x, x m R ad R m R ad f : R R beig cocave, we get: ( λ ) λ λ ( ) ( λ) ( ) h x + x h x + h x, for ay λ. Thus, h: R m R is also cocave. 6