# CH-5120: Modern Control Theory Assignment 1 solution

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## Problem 1: Change of basis (2 + 2 points)

Part-1.1: Change of basis for a vector
Let � = #

�&, where a and b are the last two digits of your roll number. Thus, if this is
CH17B987, then � = #
8
7
&. (If your roll number ends in 00, then use a = 1 and b = 1).
1. Express this vector x in terms of new basis, �! = #
1
1
& and �” = # 1
−1
&.

Part-1.2: Change of bases for linear transformation
Recall that the problem of blending of two streams was a three-input-two-output problem.
The inputs were flowrates �!, �”, �#\$% and the outputs were ℎ, �&. The gain matrix is given by:
� = # 4 2 4
0.5 1 0
&

2. How will this matrix change if the domain space is expressed in terms of the following bases:
�! = 6
1
0
0
7, �” = 6
1
1
0
7, �’ = 6
0
1
−1
7
and the co-domain space is expressed in terms of
�! = #
1
1
&, �” = # 1
−1
&

## Problem 2: Linearly Dependence (1 + 3 points)

Let �!, �”, �’ ∈ ℝ( be linearly independent vectors in n-dimensional space.
3. If the above three vectors are linearly independent, what is/are the possible value(s) of n?
(a) n = 1 (b) n = 2 (c) n = 3 (d) n = 4 (e) n = 5

4. Consider the three vectors � = �! + �”, � = �! + �’, � = �” + �’. Are the vectors u, v, w
linearly independent? Prove this.

## Problem 3: Null and Image Spaces (4 points)

Consider a matrix L = # � � 4
0.5 1 0
&, where a and b are the last two digits of your roll number.
If your roll number ends in 00, use a = 1 and b = 1.

5. Using definition, determine null space and image space of L.
Using MATLAB: Not Graded, For Practice Only
Also confirm the same using SVD (please use MATLAB for SVD).

## Problem 4: Eigenvalue Decomposition and Matrix Exponent (1 + 1 + 1 + 1 points)

Consider a matrix: � = #
1 0
� �
&, where a and b are the last two digits of your roll number.
Thus, if this is CH17B987, then � = #
1 0
8 7
&. (If your roll number ends in 00, use a = 1 and b = 1).

6. Obtain the characteristic equation and hence compute the eigenvalues of B.

7. Substitute B in its characteristic equation and thus verify Cayley Hamilton Theorem.

8. Perform eigenvalue decomposition for the matrix B

9. Using eigenvalue decomposition, compute matrix exponent �&
Using MATLAB: Not Graded, For Practice Only
Use MATLAB and compute the matrix exponent of B.

## Problem 5: Jordan Decomposition (2 + 2 points)

10. Find the eigenvalues of the matrix, � = # 1 1
−1 3
&. Since eigenvalues are repeated, compute
eigenvector and generalized eigenvector

11. Representing � = �Λ�)! in Jordan canonical form, compute the matrix exponent
Hints for Problems 4 and 5
• Please see the discussion about effect of similarity transform on exponent
• Consider the following rules
exp F#
� 0
0 �
&H = I
�* 0
0 �*J, exp F#
� �
0 �
&H = I
�* ��*
0 �* J