Question
Which of the following statements are true and which are false? Give reasons for your answer.
i) If Vis a finite dimensional vector space and 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 a is the eigenvalue of a matrix A with characteristic polynomial f(x), (x - )k | f(x) and
then the geometric multiplicity of a is at most k.
iv) If = 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 n x n matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any 2 x 3 matrix is unique.
Answer :
Word Count : 926
Let's analyze each statement carefully and determine whether it is true or false, providing justifications. --- ### (i) If \( V \) is a finite-dimensional vector space and \( T: V \to V \) is a diagonalizable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of \( T \) is diagonal. #### Answer: False Reason: If \( T \) is diagonalizable, then there exists a basis of eigenvectors in which its matrix representation is diagonal. However, this basis is not necessarily unique. If an eigenvalue has geometric multiplicity greater than 1, then we can choose different eigenvectors to form different bases. The basis is not unique, but the diagonal form (up to permutation) is unique. --- ### (ii) Up to similarity, there is a unique \( 3 \times 3 \) matrix with minimal polynomial \( (x - 1)^2 (x - 2) \). #### Answer: True Reason: The minimal polynomial of a matrix determines its Jordan form. The given polynomial suggests that the matrix has eigenvalues \( 1 \) and \( 2 \), with \( 1 \) having an algebraic multiplicity of 2 and \( 2 \) having an algebraic multiplicity of 1. The presence of \( (x - 1)^2 \) indicates a nontrivial Jordan block. The Jordan form is unique up to similarity, which means there is a unique ___ ______ _______ _____ ___ ________ _______.
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Let's analyze each statement carefully and determine whether it is true or false, providing justifications. --- ### (i) If \( V \) is a finite-dimensional vector space and \( T: V \to V \) is a diagonalizable linear operator, then there is a basis, unique up to order of the elements, with respect to which the matrix of \( T \) is diagonal. #### Answer: False Reason: If \( T \) is diagonalizable, then there exists a basis of eigenvectors in which its matrix representation is diagonal. However, this basis is not necessarily unique. If an eigenvalue has geometric multiplicity greater than 1, then we can choose different eigenvectors to form different bases. The basis is not unique, but the diagonal form (up to permutation) is unique. --- ### (ii) Up to similarity, there is a unique \( 3 \times 3 \) matrix with minimal polynomial \( (x - 1)^2 (x - 2) \). #### Answer: True Reason: The minimal polynomial of a matrix determines its Jordan form. The given polynomial suggests that the matrix has eigenvalues \( 1 \) and \( 2 \), with \( 1 \) having an algebraic multiplicity of 2 and \( 2 \) having an algebraic multiplicity of 1. The presence of \( (x - 1)^2 \) indicates a nontrivial Jordan block. The Jordan form is unique up to similarity, which means there is a unique ___ ______ _______ _____ ___ ________ _______.
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