IIT M FOUNDATION AN EXAM QDF4 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

Linear Algebra

1. Matrices and Their Properties

   • Conjugate Transpose (Hermitian): $B^H$
   • Unitary Matrix: $A^\dagger A = A A^\dagger = I$
   • Eigenvalues and Eigenvectors: $A \mathbf{v} = \alpha \mathbf{v}$
   • Diagonalization: $A = Q \Lambda Q^T$
   • Singular Value Decomposition (SVD): $A = Q_1 \Sigma Q_2^T$
   • Determinant: $\text{det}(A)$
   • Quadratic Forms: $Q(x, y, z) = \begin{bmatrix} x & y & z \end{bmatrix} M \begin{bmatrix} x \\ y \\ z \end{bmatrix}$

1. Matrix Types and Properties

   • Symmetric Matrix: $M = M^T$
   • Diagonal Matrix: All off-diagonal elements are zero.
   • Identity Matrix: Diagonal matrix with all diagonal elements equal to 1.
   • Upper Triangular Matrix: All elements below the main diagonal are zero.
   • Lower Triangular Matrix: All elements above the main diagonal are zero.
   • Singular Matrix: Determinant is zero.
   • Invertible Matrix: Determinant is non-zero.
   • Positive Definite and Semi-Definite Matrices

1. Eigenvalue and Eigenvector Equations

   • Characteristic Equation: $\det(A - \lambda I) = 0$
   • Eigenvalue Magnitude for Unitary Matrices: $|\alpha| = 1$

1. Principal Component Analysis (PCA)

   • Covariance Matrix: $C$
   • Eigenvalues of Covariance Matrix: $\lambda_1, \lambda_2$
   • Projections onto Principal Components

Optimization

1. Gradient Descent Algorithm

   • Update Rule: $x_{t+1} = x_t - \eta_t \frac{\partial f}{\partial x}$
   • Learning Rate: $\eta_t = \frac{1}{t^2}$
   • Partial Derivatives: $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}$

Geometry

1. Closest Point on a Line

   • Line Equation: $y = 2x + 1$
   • Distance Minimization: Finding the point on the line closest to a given point.

Equations

1. Characteristic Equation for Eigenvalues:

1. Eigenvalue Magnitude for Unitary Matrices:

1. Gradient Descent Update Rule:

1. Quadratic Form:

1. Distance Minimization for Closest Point on a Line: