Course Code: MMT-002

Assignment Code: MMT-002/TMA/2023

Maximum Marks: 100

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 \(T: V \rightarrow V\) 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 \(3 \times 3\) matrix with minimal polynomial \((x-1)^{2}(x-2)\).

iii) If \(\lambda\) is the eigenvalue of a matrix \(A\) with characteristic polynomial \(f(x),(x-\lambda)^{k} \mid f(x)\) and \((x-\lambda)^{k+1} \nmid f(x)\), then the geometric multiplicity of \(\lambda\) is at most \(k\).

iv) If \(\rho(A)=1\), then \(A^{k} \rightarrow \infty\) as \(k \rightarrow \infty\).

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 a \(n \times n\) matrix need not be unique.

ix) All the entries of a positive definite matrix are non-negative.

x) The SVD of any \(2 \times 3\) matrix is unique.

2) a) Let \(T: \mathbf{C}^{2} \rightarrow \mathbf{C}^{2}: T\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}x+2 y-i z \\ 2 y+i z \\ i x+z-2 z\end{array}\right]\). Find \([T]_{B},[T]_{B^{\prime}}\) and \(P\) where

\[
B=\left\{\left[\begin{array}{l}
0 \\
i \\
0
\end{array}\right],\left[\begin{array}{c}
i \\
1 \\
-1
\end{array}\right],\left[\begin{array}{l}
0 \\
0 \\
2
\end{array}\right]\right\}, B^{\prime}=\left\{\left[\begin{array}{c}
1 \\
-i \\
1
\end{array}\right],\left[\begin{array}{l}
0 \\
0 \\
1
\end{array}\right],\left[\begin{array}{l}
1 \\
i \\
0
\end{array}\right]\right\},[T]_{B^{\prime}}=P^{-1}[T]_{B} P
\]

b) If \(C\) and \(D\) are \(n \times n\) matrices such that \(C D=-D C\) 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.

c) Can \(A\) be similar to \(A+I\) ? Give reasons for your answer.

3) Find the Jordan canonical form \(J\) for

\(B=\left[\begin{array}{cccc}-1 & 0 & -2 & -4 \\ 2 & 1 & 2 & 4 \\ -4 & 2 & -1 & -4 \\ 2 & -1 & 1 & 3\end{array}\right]\)

Also, find a matrix \(P\) such that \(J=P^{-1} B P\).

4) a) 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 \(\mathrm{M}\) and 10\% of M’s population moves to T. What is the long term effect of on the population of \(\mathrm{M}\) and T? Are they likely to stabilise?

b) Solve the following system of differential equations:

\[
\frac{d y(t)}{d t}=A y(t) \text { with } y(0)=\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \text { where } A=\left[\begin{array}{ccc}
2 & -5 & -11 \\
0 & -2 & -9 \\
0 & 1 & 4
\end{array}\right]
\]

5) a) Let

\[
A=\left[\begin{array}{ccc}
2 & 2 & 1 \\
-1 & -1 & 2 \\
0 & 0 & -2
\end{array}\right]
\]

Find a unitary matrix \(U\) such that \(U^{*} A U\) is upper triangular.

b) Use least squares method to find a quadratic polynomial that fits the following data: \((-2,15.7),(-1,6.7),(0,2.7),(1,3.7),(2,9.7)\).

6) a) Check which of the following matrices is positive definite and which is positive semi-definite:

\[
A=\left[\begin{array}{lll}
1 & 1 & 0 \\
1 & 2 & 1 \\
0 & 1 & 1
\end{array}\right], B=\left[\begin{array}{ccc}
2 & 0 & 1 \\
0 & 2 & -1 \\
1 & -1 & 3
\end{array}\right]
\]

Also, find the square root of the positive definite matrix.

b) Find the \(\mathrm{QR}\) decomposition of the matrix

\[
\left[\begin{array}{ccc}
2 & -2 & 1 \\
2 & 2 & 1 \\
0 & 1 & 1 \\
1 & 0 & 1
\end{array}\right]
\]

7) Find the SVD of the following matrices:
i) \(\left[\begin{array}{ccc}-1 & 1 & 1 \\ 1 & 1 & 0\end{array}\right]\)
ii) \(\left[\begin{array}{cc}-1 & 1 \\ 1 & 1 \\ 1 & 2\end{array}\right]\)