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IIT M DIPLOMA AN EXAM QDD4 16 Mar - 16 Mar 2025

MLF Quiz 214 questions40 marks9 MCQ · 3 MSQ · 2 SADiploma / afternoon / QDD4

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Q1MCQ3 marks

Q2MCQ3 marks

Q3MCQ3 marks

Q4MCQ2 marks

Q5MSQ3 marksselect all that apply

Q6MSQ3 marksselect all that apply

Which of the following statements is/are True regarding Schur’s Theorem?

Every square matrix is unitarily similar to an upper triangular matrix.
The diagonal entries of the Schur form of a matrix are its eigenvalues.
A square matrix with repeated eigenvalues cannot always be unitarily similarto an upper triangular matrix.
Schur’s Theorem holds only for square matrices, as it requires unitarysimilarity and the presence of eigenvalues.

Q7MSQ3 marksselect all that apply

Q8SA3 marks

Your numeric answer:

Q9SA3 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q10MCQ2 marks

Compute the sample covariance matrix C for the given dataset.

Q11MCQ3 marks

What is the principal direction chosen for PCA?

Q12MCQ3 marks

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q13MCQ3 marks

Which of the following options is/are true for Q1?

Q14MCQ3 marks

Answer key

Q1 (MCQ, 3 marks): C
Q2 (MCQ, 3 marks): B
Q3 (MCQ, 3 marks): D
Q4 (MCQ, 2 marks): A
Q5 (MSQ, 3 marks): C, D
Q6 (MSQ, 3 marks): A, B, D
Q7 (MSQ, 3 marks): A, D
Q8 (SA, 3 marks): 2.41 to 2.47
Q9 (SA, 3 marks): -1
Q10 (MCQ, 2 marks): C
Q11 (MCQ, 3 marks): B
Q12 (MCQ, 3 marks): A
Q13 (MCQ, 3 marks): D
Q14 (MCQ, 3 marks): D

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Summary of Core Topics, Concepts, Principles, and Equations

1. Taylor Series Expansion

Concept: Approximating a function using an infinite sum of terms calculated from the values of its derivatives at a single point.
Equation: For $f(x) = \frac{1}{x}$ around $x = 1$ :

f(x) \approx 1 - (x-1) + (x-1)^2 - (x-1)^3 + (x-1)^4

2. Eigenvalues and Eigenvectors

Concept: Eigenvalues are scalars associated with a linear transformation of a vector space, and eigenvectors are the corresponding vectors.
Equation: For a Hermitian matrix $A$ , the eigenvalues $\lambda$ satisfy $\det(A - \lambda I) = 0$ .

3. Schur's Theorem

Concept: Every square matrix is unitarily similar to an upper triangular matrix.
Principle: The diagonal entries of the Schur form are the eigenvalues of the matrix.

4. Gradient Descent

Concept: An optimization algorithm used to minimize some function by iteratively moving in the direction of steepest descent.
Equation: Update rules for gradient descent:

x_{new} = x_{old} - \eta \frac{\partial f}{\partial x}

y_{new} = y_{old} - \eta \frac{\partial f}{\partial y}

where $\eta$ is the learning rate.

5. Singular Value Decomposition (SVD)

Concept: A factorization of a real or complex matrix, with applications in signal processing and statistics.
Equation: For a matrix $M$ , SVD is given by:

M = Q_1 \Sigma Q_2^T

where $Q_1$ and $Q_2$ are unitary matrices, and $\Sigma$ is a diagonal matrix with singular values.

6. Principal Component Analysis (PCA)

Concept: A statistical procedure that uses orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables.
Equation: The covariance matrix $C$ is given by:

C = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(x_i - \bar{x})^T

where $n$ is the number of data points, $x_i$ are the data points, and $\bar{x}$ is the mean of the data points.

7. Positive Definite Matrices

Concept: A matrix is positive definite if it satisfies certain conditions related to its eigenvalues and quadratic forms.
Principle: All eigenvalues of a positive definite matrix are positive.