IIT M FOUNDATION AN EXAM QDF2 01 Dec - 01 Dec 2024

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. Matrix Properties and Operations

• Conjugate Transpose (Hermitian Transpose): $B^H$
• Eigenvalues and Eigenvectors: $A \mathbf{v} = \lambda \mathbf{v}$
• Diagonalization: $A = Q \Lambda Q^T$
• Singular Value Decomposition (SVD): $A = Q_1 \Sigma Q_2^T$
• Matrix Types: Symmetric, Diagonal, Upper Triangular, Lower Triangular, Singular, Identity

2. Matrix Equations and Properties

• Determinant: $\text{det}(A)$
• Trace: $\text{tr}(A)$
• Invertibility: $A$ is invertible if $\text{det}(A) \neq 0$
• Unitary Matrix: $A^H A = I$
• Hermitian Matrix: $A = A^H$

3. Quadratic Forms and Covariance Matrices

• Quadratic Form: $Q(x, y, z) = \mathbf{x}^T M \mathbf{x}$
• Covariance Matrix: $C = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})(x_i - \bar{x})^T$

4. Principal Component Analysis (PCA)

• Eigenvalues of Covariance Matrix: $\lambda_1, \lambda_2$
• Projections onto Principal Components: $\tilde{x}_i$

5. Gradient Descent Algorithm

• Gradient Descent Update Rule: $\theta_{t+1} = \theta_t - \eta \nabla f(\theta_t)$
• Learning Rate: $\eta = \frac{1}{t^2}$

6. Optimization and Distance Minimization

• Closest Point on a Line: Minimize the distance from a point to a line $y = 2x + 1$

Important Equations

Matrix Operations

1. Eigenvalue Equation:

1. Diagonalization:

1. Singular Value Decomposition (SVD):

1. Quadratic Form:

1. Covariance Matrix:

1. Gradient Descent Update Rule:

1. Distance from a Point to a Line: