Problem 1E: For each of the following pairs of vectors z and w, compute (i) z, (ii) w, (iii) z,w, and (iv) w,z:... Problem 2E: Let z1=( 1+i 2 1i 2 ) and z2=( i 2 1 2 ) Show that {z1,z2} is an orthonormal set in 2. Write the... Problem 3E: Let {u1,u2} be an orthonormal basis for 2, and let z=(4+2i)u1+(65i)u2. What are the values of... Problem 4E: Which of the matrices that follow are Hermitian? Normal? ( 1i223) (1 2i 2+i 1) ( 1 2 1 2 1 2 1 2 )... Problem 5E: Find an orthogonal or unitary diagonalizing matrix for each of the following: (2112) (1 3+i 3i4)... Problem 6E Problem 7E: Let A be an nn Hermitian matrix and let x be a vector in n. Show that if c=xHAx, then c is real. Problem 8E: Let A be an Hermitian matrix and let B=iA. Show that B is skew Hermitian. Problem 9E: Let A and C be matrices in mn and let Bnr. Prove each of the following rules: (AH)H=A (A+C)H=AH+CH... Problem 10E Problem 11E: Show that z,w=wHz defines an inner product on n. Problem 12E: Let x, y, and z be vectors in n and let and be complex scalars. Show that z,x+y=z,x+z,y Problem 13E: Let {u1,...,un} be an orthonormal basis for a complex inner product space V, and let... Problem 14E: Given that A=(40001i0 i1) find a matrix B such that BHB=A. Problem 15E: Let U be a unitary matrix. Prove that U is normal. Ux=x forall xn. if is an eigenvalue of U, then... Problem 16E: Let u be a unit vector in n and define U=I2uuH. Show that U is both unitary and Hermitian and,... Problem 17E: Show that if a matrix U is both unitary and Hermitian then any eigenvalue of U must equal either 1... Problem 18E: Let A be a 22 matrix with Schur decomposition UTUH and suppose that t120. Show that the eigenvalues... Problem 19E: Let A be a 55 matrix with real entries. Let A=QTQT be the real Schur decomposition of A, where T is... Problem 20E: Let A be a nn matrix with Schur decomposition UTUH. Show that if the diagonal entries of T are all... Problem 21E: Show that M=A+iB (where A and B are real matrices) is skew Hermitian if and only if A is skew... Problem 22E Problem 23E Problem 24E Problem 25E Problem 26E: Let A be a Hermitian matrix with eigenvalues 1,...,n and orthonormal eigenvectors u1,...,un. Show... Problem 27E: Let A=(0110) Write A as a sum 1u1u1T+2u2u2T, where 1 and 2 are eigenvalues and u1 and u2 are... Problem 28E: Let A be a Hermitian matrix with eigenvalues 12n and orthonormal eigenvectors u1,...,un. For any... Problem 29E: Given Amm,Bnn,Cmn, the equation AXXB=C(3) is known as Sylvester's equation. An mn matrix X is said... format_list_bulleted