IGNOU MMT 2 SOLVED ASSIGNMENT

Ques 1.

 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, e^N 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.

Ques 2.

 Let . Find [T]_B, [T]_B' and P where 
.

Ques 3.

 If C and D are  matrices such that  and D^-1 exists, then show that C is similar to -C. Hence show that the eigenvalues of C must come in plus-minus pairs. 

Ques 4.

 Find the Jordan canonical form J for

.
 Also, find a matrix P such that . 

.

Ques 5.

 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? 

Ques 6.

 Solve the following system of differential equations: 
 with , where 

Ques 7.

 Let

Find a unitary matrix U such that U^*AU is upper triangular. 

Ques 8.

 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).

Ques 9.

 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. 

Ques 10.

Find the QR decomposition of the matrix 

Ques 11.

 Find the SVD of the following matrices: 

i) 

ii) 

Ques 12.

 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, e^N 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.

Ques 13.

 Let . Find [T]_B, [T]_B' and P where 
.

Ques 14.

 If C and D are  matrices such that  and D^-1 exists, then show that C is similar to -C. Hence show that the eigenvalues of C must come in plus-minus pairs. 

Ques 15.

 Find the Jordan canonical form J for

.
 Also, find a matrix P such that . 

.

Ques 16.

 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? 

Ques 17.

 Solve the following system of differential equations: 
 with , where 

Ques 18.

 Let

Find a unitary matrix U such that U^*AU is upper triangular. 

Ques 19.

 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).

Ques 20.

 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. 

Ques 21.

Find the QR decomposition of the matrix 

Ques 22.

 Find the SVD of the following matrices: 

i) 

ii) 

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