Problems - Lecture 2

Linear Algebra for Wireless Communications

1. Assume that all pivot variables come last when performing Gauss-Jordan elim- ination of an M N matrix. Describe the four blocks in the resulting M N reduced row echelon form (the block B is of size r r):   AB R = : CD   2. Rank of products

(a) Show that rank (AB) min rank A; rank B unless B is nonsingular, in which case rank (AB) = rankf (A). Hint: Theg columns of AB are linear combinations of the columns of A. (b) If A is M N, M > N and of full rank, prove that AT A is nonsingular.  (c) Show Sylvester’s inequality which says that rank A + rank B N rank (AB) min rank A; rank B if A is M N and B is N P .   f g   3. Matrices of rank r

(a) If u is N 1 and v is M 1, show that the matrix uvT has rank 1. Show, conversely, that any matrix of rank 1 can be expressed as the product of a column matrix and a row matrix. (b) Show that in general rank (A + B) rank A + rank B.  (c) Show that a matrix A has rank r if an only if it can be written in the form r T A = uivi 1 X where ui and vi are sets of independent column vectors. f g f g 4. Toeplitz matrices can be used to represent polynomial multiplication.

(a) Show that if a (s) b (s) = c (s), with a (s) of degree N and b (s) of degree M, then T (a) b = c; where (in an obvious notation)

T T b = [b0 : : : bM ] , c = [c0 : : : cM+N ]

1 and T (a) is an (M + N + 1) (M + 1) lower triangular Toeplitz matrix de…ned by its …rst column 

T [a0 : : : aN 0 ::: 0] :

(b) Show that we can write c = T (b) a where T (b) is an (M + N + 1) (N + 1) lower triangular Toeplitz matrix de…ned by its …rst column 

T [b0 : : : bM 0 ::: 0] :

(c) What can you say about the commutativity of Toeplitz matrices from the fact that polynomials commute?

5. Polynomial …tting where the coe¢ cients pk of a polynomial p (s) = p0 + p1s + N 1 ::: + pN 1s are selected so that the polynomial passes through a set of N given points p ( k) = ck (k = 0; 1;:::;N 1) can be done by solving Vp = c

where V is Vandermonde matrix whose ith row is

N 1 1 i ;


i the vector   T p = p0 p1 pN 1


contains the unknown polynomial coe¢ cients, and the vector

T c = c0 c1 cN 1


contains the N known values of the polynomial (in the correct order). By direct evaluation (or by row operations), show that

(a) the 3 3 Vandermonde determinant is given by ( 3 1)( 3 2)( 2 1),  (b) the N N Vandermonde determinant is equal to ( j i) where the sum is taken over (i; j) such that 0 i < j N 1, and   Q (c) form this describe how to extract the restrictions on when we can …t an Nth degree polynomial through N known points (as if you didn’t know already), i.e., the restrictions under which the system Vp = c can be solved.

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