IIT M FOUNDATION AN EXAM QDF3 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

   • Hermitian Matrix: A matrix $B$ is Hermitian if $B = B^H$, where $B^H$ is the conjugate transpose of $B$.
   • Unitary Matrix: A matrix $A$ is unitary if $A^\dagger A = A A^\dagger = I$, where $A^\dagger$ is the conjugate transpose of $A$ and $I$ is the identity matrix.
   • Determinant and Trace: The determinant of a matrix $B$ is zero if the matrix is singular. The trace of a matrix is the sum of its diagonal elements.
   • Eigenvalues and Eigenvectors: For a unitary matrix, the magnitude of the eigenvalues is 1, and the eigenvectors corresponding to different eigenvalues are orthogonal.
   • Diagonalization: A matrix $A$ can be diagonalized as $A = Q \Lambda Q^T$, where $Q$ is a matrix of eigenvectors and $\Lambda$ is a diagonal matrix of eigenvalues.

1. Quadratic Forms

   • Matrix Representation: A quadratic form $Q(x, y, z)$ can be represented in matrix form as $Q(x, y, z) = \begin{bmatrix} x & y & z \end{bmatrix} M \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, where $M$ is a symmetric matrix.

1. Gradient Descent

   • Update Rule: For each iteration $t$, the update rule is:

• Learning Rate: The learning rate $\eta_t$ can be defined as $\frac{1}{t^2}$.

1. Singular Value Decomposition (SVD)

   • SVD Form: A matrix $A$ can be written in SVD form as $A = Q_1 \Sigma Q_2^T$, where $Q_1$ and $Q_2$ are unitary matrices, and $\Sigma$ is a diagonal matrix of singular values.

1. Principal Component Analysis (PCA)

   • Covariance Matrix: The sample covariance matrix $C$ for a dataset is calculated as:

• Eigenvalues and Eigenvectors: The eigenvalues of the covariance matrix are used to determine the principal components.

Equations

1. Eigenvalue Equation:

where $A$ is a matrix, $v$ is an eigenvector, and $\lambda$ is an eigenvalue.

1. Gradient Descent Update:

where $\eta_t$ is the learning rate and $\nabla f(x_t)$ is the gradient of the function $f$ at $x_t$.

1. SVD Equation:

where $Q_1$ and $Q_2$ are unitary matrices, and $\Sigma$ is a diagonal matrix of singular values.

1. Covariance Matrix:

where $n$ is the number of data points, $x_i$ are the data points, and $\bar{x}$ is the mean of the data points.