Which of the following statements are true and which are false? Give reasons for your answer.
i) If v is a finite dimensional vector space and T:v→ v is a diagonalisable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of T is diagonal.
ii) Up to similarity, there is a unique 3 x 3 matrix with minimal polynomial (x- 1)^2
(x-2).
iii) If λ is the eigenvalue of a matrix A with characteristic polynomial f(x), (x-λ)(x-λ)^k
if (x) and (x-λ)k+1 + f(x), then the geometric multiplicity of λ is at most k.
iv) If p(A) = 1, then Ak → ∞ as k → ∞.
v) If N is nilpotent, eN is also nilpotent.
vi) The sum of two normal matrices of the order n is normal.
vii) If P and Q are positive definite operators, P+Q is a positive definite operator.
viii) Generalised inverse of a nxn matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any 2 × 3 matrix is unique.

i) True. Since T is diagonalizable, there exists a basis of eigenvectors of T. In this basis, the matrix of T is diagonal.

ii) False. The minimal polynomial determines the possible Jordan canonical forms of the matrix, but it does not uniquely determine the matrix itself.

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