IIT M DIPLOMA AN EXAM QDD2 16 Mar - 16 Mar 2025

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

Mathematics

1. 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: $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$

1. Eigenvalues and Eigenvectors

   • Concept: Eigenvalues are scalars associated with a linear transformation of a vector space, and eigenvectors are the corresponding vectors that are scaled by the transformation.
   • Equation: $A v = \lambda v$, where $A$ is a matrix, $v$ is an eigenvector, and $\lambda$ is an eigenvalue.

1. Hermitian Matrix

   • Concept: A square matrix that is equal to its own conjugate transpose.
   • Property: Eigenvalues of a Hermitian matrix are real.

1. Schur's Theorem

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

1. Singular Value Decomposition (SVD)

   • Concept: Factorizing a matrix into three matrices $U$, $\Sigma$, and $V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix.
   • Equation: $A = U \Sigma V^T$

1. Principal Component Analysis (PCA)

   • Concept: A technique for reducing the dimensionality of a dataset by projecting it onto a lower-dimensional subspace.
   • Steps: Compute the covariance matrix, find the eigenvalues and eigenvectors, and project the data onto the principal components.

1. Gradient Descent

   • Concept: An optimization algorithm for finding the minimum of a function.
   • Equation: $x_{n+1} = x_n - \gamma \nabla f(x_n)$, where $\gamma$ is the learning rate and $\nabla f(x_n)$ is the gradient of the function at $x_n$.

Data Science

1. Covariance Matrix

   • Concept: A matrix that contains the covariances associated with several variables.
   • Equation: $C_{ij} = \text{Cov}(X_i, X_j)$

1. Projection onto Principal Components

   • Concept: Projecting data onto the directions of maximum variance.
   • Equation: $\tilde{x} = x^T v$, where $v$ is the eigenvector corresponding to the largest eigenvalue.