Solve the following system of differential equations: with
, where
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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 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 matrix with minimal polynomial (x - 1)2(x - 2).
iii) If is the eigenvalue of a matrix A with characteristic polynomial f(x),
and
, then the geometric multiplicity of
is at most k.
iv) If , then
as
.
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 an matrix need not be unique.
ix) All the entries of a positive definite matrix are non-negative.
x) The SVD of any matrix is unique.
Check which of the following matrices is positive definite and which is positive semi-definite:
Also, find the square root of the positive definite matrix.
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 of T's population moves to M and
of M's population moves to T. What is the long term effect on the population of M and T? Are they likely to stabilise?
Use least squares method to find a quadratic polynomial that fits the following data:
(-2, 15.7), (-1, 6.7), (0, 2.7), (1, 3.7), (2, 9.7).
Solve the following system of differential equations: with
, where
Find the QR decomposition of the matrix
Find the SVD of the following matrices:
i)
ii)