Q1. Attempt all parts of Question 1.
- (a) i)) Define skew-hermitian matrix, orthogonal matrix and unitary matrix with an example of each. (150 words)
- (a) ii)) Determine the eigenvalues and eigenvectors of the following matrix A: A = [[1, 1, 1], [1, 2, 1], [3, 2, 3]] (150 words)
- (b)) Show that every eigenvalue of a unitary matrix is of unit modulus. (250 words)
- (c)) Identify the conic section whose equation is 5x^2 - 4xy + 5y^2 = 4 (250 words)
- (d)) Define covariant and contravariant tensors of rank two. Show that the velocity and acceleration are contravariant vectors. (250 words)
- (e)) Show that the roots of the equation z^4 - 1 = 0 form a cyclic group of order 4. (250 words)
Key Points:
- Skew-Hermitian matrix: A† = -A (conjugate transpose equals negative).
- Orthogonal matrix: OᵀO = I (transpose equals inverse, for real matrices).
- Unitary matrix: U†U = I (conjugate transpose equals inverse, for complex matrices).
- Eigenvalues of unitary matrices always have unit modulus (|λ|=1).
Answer: