IIT M FOUNDATION AN EXAM QDF2 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

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 $x = 1$:

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 of a matrix are its eigenvalues.

4. Singular Value Decomposition (SVD)

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

5. Gradient Descent

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

where $\eta$ is the learning rate.

6. Principal Component Analysis (PCA)

• Concept: A technique used to emphasize variation and bring out strong patterns in a dataset.
• Equation: Covariance matrix $C$ for a dataset $X$:

7. Positive Definite Matrices

• Concept: A matrix is positive definite if it satisfies $x^T A x > 0$ for all non-zero vectors $x$.
• Principle: All singular values of a positive definite matrix are positive.

8. Unitary Matrices

• Concept: A matrix $U$ is unitary if $U^* U = I$.
• Principle: The inverse of a unitary matrix is its conjugate transpose.