IIT M DIPLOMA AN EXAM QDD3 03 Aug - 03 Aug 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

1. Optimization and Critical Points

• Gradient and Hessian Matrix: For a function $f: \mathbb{R}^n \to \mathbb{R}$, the gradient $\nabla f(x^*) = 0$ and the Hessian matrix $\nabla^2 f(x^*)$ being positive definite indicate a local minimum.
• Quadratic Forms: Understanding positive definiteness, positive semi-definiteness, negative definiteness, and negative semi-definiteness of quadratic forms.

2. Matrix Properties

• Positive Definite Matrices: A matrix $A$ is positive definite if $x^T A x > 0$ for all non-zero vectors $x$.
• Eigenvalues and Eigenvectors: For a Hermitian matrix, eigenvalues are real, and eigenvectors corresponding to different eigenvalues are orthogonal.
• Singular Value Decomposition (SVD): For a matrix $A$, SVD is given by $A = Q_1 \Sigma Q_2^T$, where $Q_1$ and $Q_2$ are orthogonal matrices, and $\Sigma$ is a diagonal matrix of singular values.

3. Principal Component Analysis (PCA)

• Covariance Matrix: The sample covariance matrix $C$ is used to find the principal components.
• Eigenvalues and Principal Components: The eigenvalues of the covariance matrix represent the variance in the direction of the corresponding eigenvectors (principal components).

4. Gradient Descent

• Gradient Descent Algorithm: Iterative method to minimize a function $f(x, y)$ using the update rule $x_{t+1} = x_t - \eta_t \nabla f(x_t)$, where $\eta_t$ is the learning rate.

5. Distance Minimization

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Equations

1. Gradient and Hessian:

1. Quadratic Form:

1. Positive Definite Matrix:

1. Eigenvalues of $AA^T$ and $A^T A$:

1. Principal Component Direction:

1. Gradient Descent Update:

1. Distance Formula:

1. Sample Covariance Matrix: