Can A be similar to A+I? Give reasons for your answer.
No, A cannot be similar to A + I.
Let λ be an eigenvalue of A, and let x be the corresponding eigenvector, i.e., Ax = λx. Then, if A were similar to A + I, there would exist an __________ _________ _________ _______ _________ _________ ________ _______ ___ ___ ___ _____.
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Let M and T be a metro city and a nearby district town, respectively. Our government is trying to develop infrastructure in T so that people shift to T. Each year 15% of T’s population moves to M and 10% of M’s population moves to T. What is the long term effect of on the population of M and T? Are they likely to stabilise?
If C and D are nxn matrices such that CD = −DC and D^{-1} exists, then show that C is similar to -D. Hence show that the eigenvalues of C must come in plus-minus pairs.
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.
Find the Jordan canonical form J for
Also, find a matrix P such that
Can A be similar to A+I? Give reasons for your answer.
Solve the following system of differential equations:
where