problem1) Which of the following statements are true? Give reasons for your answer.
i) If A and B are similar matrices, they must be congruent.
ii) If A is a positive definite matrix, then so is A^{-1}
iii) If A and B are n X n similar matrices, then they have the same eigenspaces.
iv) If T is a diagonalisable linear operator, then the geometric multiplicity of each of its eigenvalues is 1.
v) If T has a unique Jordan form J, then J must be a diagonal matrix.
vi) The spectral radius of a square matrix has to be a positive real number.
vii) ∃ A ∈ M_{3} (C) such that A is not triangulisable.
viii) If the row sum of a square matrix A is d, then d is an eigenvalue of A.
ix) If B is the Moore-Penrose inverse of A, then A is t he Moore-Penrose inverse of B.
problem2a) Let T:C^{3}→ C^{3} T: Fin[T]_{B}, [T]_{B'}and P , where
b) If C and D are n × n matrices such that CD =- DC and D^{-1}exists, then show that C is similar to -C. Hence show that the eigenvalues of C must come in plus-minus pairs. Further, give an ex of a square matrix A which is not similar to –A.
c) Can A be similar to A+I? Give reasons for your answer.
problem3)a) Let T: R^{3}→ R^{3}:. Obtain the generalised eigenspaces of T of orders 2 and 3, for each eigenvalue of T.
b) Find the possible Jordan forms of B
c) Let. Find an invertible matrix P so that P^{-1} AP is in block diagonal form.