IIT M FOUNDATION AN EXAM QDF3 16 Mar - 16 Mar 2025

MLF Quiz 214 questions40 marks9 MCQ · 3 MSQ · 2 SAFoundation / afternoon / QDF3

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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.

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Your numeric answer:

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Your numeric answer:

Comprehension passage - the questions below it refer to this

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

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Compute the sample covariance matrix C for the given dataset.

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What is the principal direction chosen for PCA?

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Comprehension passage - the questions below it refer to this

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

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

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Answer key

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

Taylor Series Expansion
- Concept: Representing a function as 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

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

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.

Singular Value Decomposition (SVD)
- Concept: Factorizing a matrix into three matrices $U$ , $\Sigma$ , and $V^T$ .
- Equation: $M = U \Sigma V^T$

Gradient Descent
- Concept: An optimization algorithm used to minimize some function by iteratively moving in the direction of steepest descent.
- Equation: Update rule for $x$ and $y$ :

x_{new} = x_0 - \alpha \frac{\partial f}{\partial x}

y_{new} = y_0 - \alpha \frac{\partial f}{\partial y}

Principal Component Analysis (PCA)
- Concept: A technique used to emphasize variation and bring out strong patterns in a dataset.
- Equation: Projection of data onto the principal components.

Covariance Matrix
- Concept: A matrix of covariance between pairs of variables in a dataset.
- Equation: For a dataset $X$ , the covariance matrix $C$ is given by:

C = \frac{1}{n-1} X^T X

PCA Projections
- Concept: Projecting data onto the principal components to reduce dimensionality.
- Equation: Projection of a data point $x$ onto a principal component $v$ :

\text{proj}_v x = \frac{x \cdot v}{v \cdot v} v

Matrix Operations
- Concept: Basic operations on matrices such as addition, multiplication, and decomposition.
- Principle: Understanding and implementing matrix operations efficiently.