Q1. 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. (70 words)
- (ii)) Up to similarity, there is a unique 3 × 3 matrix with minimal polynomial (x – 1)²(x – 2). (70 words)
- (iii)) If λ is the eigenvalue of a matrix A with characteristic polynomial f(x), (x − λ)ᵏ | f(x) and (x − λ)ᵏ⁺¹ + f(x), then the geometric multiplicity of λ is at most k. (70 words)
- (iv)) If ρ(A) = 1, then Aᵏ → ∞ as k → ∞. (70 words)
- (v)) If N is nilpotent, eᴺ is also nilpotent. (70 words)
- (vi)) The sum of two normal matrices of the order n is normal. (70 words)
- (vii)) If P and Q are positive definite operators, P + Q is a positive definite operator. (70 words)
- (viii)) Generalised inverse of a n × n matrix need not be unique. (70 words)
- (ix)) All the entries of a positive definite matrix are non-negative. (70 words)
- (x)) The SVD of any 2 × 3 matrix is unique. (70 words)
Key Points:
- Diagonalizable operators have a basis of eigenvectors, but it's not unique due to repeated eigenvalues.
- Minimal polynomial uniquely determines Jordan Canonical Form (JCF) for a given matrix size, defining similarity.
- Geometric multiplicity of an eigenvalue is always less than or equal to its algebraic multiplicity.
- Spectral radius ρ(A)=1 does not guarantee Aᵏ diverges; it can converge or oscillate.
Answer: This response evaluates ten statements related to Linear Algebra concepts such as diagonalisability, minimal polynomials, eigenvalues, spectral radius, nilpotent matrices, normal matrices, positive definite operators, generalised inverses, and Singular Value Decomposition (SVD). Each statement is assessed for its truth value, and a concise reason is provided, drawing upon fundamental definitions and theorems typically covered in an MMT-002 course. The analysis emphasizes common misconceptions an...