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).
Let the required quadratic polynomial be ( y = ax^2 + bx + c ). For the data points ((-2,15.7), (-1,6.7), (0,2.7), (1,3.7), (2,9.7)), compute the needed sums: [ \sum __________ ___ __________ ________ __________ ______.
___ _______ ______ ______ ______ ___ ___ _________ ________.
____ ____ ___ _________ ____ _____ ________ _______ _____ _______.
__________ _________ _________ ______ ____ ______ ____ _____ __________ _____ _________.
________ _____ _____ _________ _____ __________ ______ _______.
_________ ___ ___ _______ ________ _____ __________ _______ _______.
________ _____ __________ ______ ___ ______ _____ _______ ________ __________ ____ ________.
__________ _______ ____ ___ _______ ____ _________ ___ _______ ________ _________.
_________ ______ ______ _________ _________ ______ ________ ________ _______ ______.
___ ______ _______ __________ ___ ________ ___ ___ _________ ______.
________ ___ ________ ____ _______ _____ ___ ___ ____ _____ ___ ________.
________ _____ ____ ________ _____ ____ ___.
________ _______ ____ _________ _________ ________ _______ ____ ___.
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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).
Find the Jordan canonical form J for
.
Also, find a matrix P such that
.
.
Find the SVD of the following matrices:
i)
ii)
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.
Let . Find [T]B, [T]B' and P where
.