9. Prove that, if there exist an Hadamard matrix of order n., then there exist an...
Hadamard Matrix 1). Is H12 orthogonal? and if so why?
Compute the determinant of the Hadamard matrix (which is also the volume of a hypercube in R4):
Let A be an mx n matrix and B be an n xp matrix. (a) Prove that rank(AB) S rank(A). (b) Prove that rank(AB) < rank(B).
T3. The French mathematician Jacques Hadamard (1865–1963) proved that if A is an n x n matrix each of whose entries satisfies the condition lajj| < M, then det(A) < Vn"M" (Hadamard's inequality). For the following matrix A, use this re- sult to find an interval of possible values for det(A), and then use your technology utility to show that the value of det(A) falls within this interval. 50.3 -2.4 -1.7 2.57 [0.2 -0.3 -1.2 1.4 |2.5 2.3 0.0 1.8...
9. An n × n matrix A is called nilpotent if for-one non, negalivew m, we have Ao, If A is a nilpotent matrix prov conider invertible matrix. To prove this tell me what is (1 + AY first the case where m2 and in this case show th This should help you to see how to prove the general n x n identity matrix). that 1+ As an Hin at (1+A)---A) case. (I is the 9. An n ×...
Define a Hadamard code generated from a 4 x 4 matrix, then calculate the corresponding generator and parity-check matrices. 11.5 (T):
Does the following limit exist? Prove your result. lim tan - 1- 0 Estimate the following limit: 3 2n - 1)2n + 1) n=0 rove the Convergence/Divergence of the following
2. Let A be an n x n matrix with AT =-A (a) Prove that A has value 0. (b) Prove that A has determinant 0 if n is odd.
*Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1 *Let . ., A, denote the eigenvalues of an n x n matrix A. Prove that the Frobenius 5. norm of A satisfies ΑIFΣ. i=1
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...