ℹ️Information Theory Unit 3 – Linear Algebra Foundations for Info Theory

Key Concepts and Definitions

• Linear algebra studies vector spaces, linear transformations, and matrices
• Vectors represent quantities with both magnitude and direction (velocity, force)
• Scalars represent quantities with only magnitude (temperature, speed)
  • Multiplying a vector by a scalar changes its magnitude but not its direction
• Vector spaces consist of a set of vectors that can be added together and multiplied by scalars
  • Must satisfy certain axioms (closure, associativity, commutativity)
• Basis vectors span the entire vector space through linear combinations
  • Linearly independent basis vectors cannot be expressed as a linear combination of other basis vectors
• Inner product measures the similarity between two vectors
  • Dot product is a common inner product defined as $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n$
• Orthogonal vectors have an inner product of zero and are perpendicular to each other

Vector Spaces and Subspaces

• Vector spaces are sets closed under vector addition and scalar multiplication
• Subspaces are subsets of a vector space that form a vector space themselves
  • Must contain the zero vector and be closed under vector addition and scalar multiplication
• Column space of a matrix is the subspace spanned by its column vectors
• Null space (kernel) of a matrix is the subspace of vectors that yield the zero vector when multiplied by the matrix
  • Solutions to the equation $A\mathbf{x} = \mathbf{0}$
• Row space of a matrix is the subspace spanned by its row vectors
• Rank of a matrix is the dimension of its column space or row space
  • Maximum number of linearly independent columns or rows
• Nullity of a matrix is the dimension of its null space
• Rank-nullity theorem states that rank + nullity = number of columns

Linear Transformations

• Linear transformations map vectors from one vector space to another while preserving vector addition and scalar multiplication
  • $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = cT(\mathbf{u})$
• Matrix-vector multiplication represents a linear transformation
  • Columns of the matrix represent the transformed basis vectors
• Composition of linear transformations is also a linear transformation
  • Corresponds to matrix multiplication
• Invertible linear transformations have a unique inverse that "undoes" the transformation
  • Corresponds to matrix inversion
• Kernel (null space) of a linear transformation is the set of vectors that map to the zero vector
• Range (image) of a linear transformation is the set of all possible output vectors

Eigenvalues and Eigenvectors

• Eigenvectors of a linear transformation (or matrix) are non-zero vectors that remain parallel to their original direction after the transformation
  • $A\mathbf{v} = \lambda\mathbf{v}$, where $\lambda$ is the eigenvalue
• Eigenvalues represent the scaling factor of the eigenvectors under the transformation
• Characteristic equation $\det(A - \lambda I) = 0$ determines the eigenvalues
  • Eigenvalues are the roots of the characteristic polynomial
• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• Diagonalizable matrices have a full set of linearly independent eigenvectors
  • Can be decomposed as $A = PDP^{-1}$, where $D$ is a diagonal matrix of eigenvalues and $P$ contains the eigenvectors as columns
• Eigendecomposition allows for efficient computation of matrix powers and exponentials

Matrix Operations and Properties

• Matrix addition and subtraction are element-wise operations
• Matrix multiplication is a binary operation that produces a new matrix
  • $(AB)_{ij} = \sum_{k} A_{ik}B_{kj}$
  • Associative but not commutative in general
• Identity matrix $I$ has ones on the main diagonal and zeros elsewhere
  • $AI = IA = A$ for any matrix $A$
• Inverse of a square matrix $A$ is denoted as $A^{-1}$, satisfying $AA^{-1} = A^{-1}A = I$
  • Not all matrices have inverses (singular matrices)
• Transpose of a matrix $A$ is denoted as $A^T$, obtained by interchanging rows and columns
• Symmetric matrices satisfy $A = A^T$
  • Have real eigenvalues and orthogonal eigenvectors
• Orthogonal matrices have orthonormal columns and rows
  • Satisfy $A^TA = AA^T = I$ and $A^{-1} = A^T$
• Positive definite matrices have positive eigenvalues and satisfy $\mathbf{x}^TA\mathbf{x} > 0$ for all non-zero vectors $\mathbf{x}$

Applications to Information Theory

• Singular Value Decomposition (SVD) factorizes a matrix into $A = U\Sigma V^T$
  • $U$ and $V$ are orthogonal matrices, and $\Sigma$ is a diagonal matrix of singular values
  • Used for dimensionality reduction, noise reduction, and feature extraction
• Principal Component Analysis (PCA) finds the principal components (directions of maximum variance) of a dataset
  • Eigenvectors of the covariance matrix correspond to the principal components
  • Used for data compression, visualization, and preprocessing
• Least squares approximation finds the best-fit solution to an overdetermined system of linear equations
  • Minimizes the sum of squared residuals $\|\mathbf{b} - A\mathbf{x}\|^2$
  • Solution is given by $\mathbf{x} = (A^TA)^{-1}A^T\mathbf{b}$
• Markov chains model systems transitioning between states with fixed probabilities
  • Transition matrix contains the probabilities of moving from one state to another
  • Stationary distribution is an eigenvector of the transition matrix with eigenvalue 1
• Error-correcting codes use linear transformations to add redundancy and protect against noise
  • Generator matrix maps messages to codewords, and parity-check matrix detects and corrects errors

Problem-Solving Techniques

• Gaussian elimination reduces a matrix to row echelon form
  • Used to solve systems of linear equations, find matrix inverses, and compute determinants
• Gram-Schmidt process orthonormalizes a set of vectors
  • Produces an orthonormal basis for a vector space
• Eigenvalue decomposition diagonalizes a matrix using its eigenvectors
  • Simplifies matrix powers, exponentials, and differential equations
• Singular Value Decomposition (SVD) provides a generalized eigendecomposition for non-square matrices
  • Reveals the structure and rank of a matrix
• Least squares approximation finds the best-fit solution to an overdetermined system
  • Can be solved using the normal equations or QR decomposition
• Spectral theorem states that symmetric matrices have an orthonormal basis of eigenvectors
  • Allows for efficient computation and analysis of symmetric matrices
• Matrix factorizations (LU, QR, Cholesky) decompose a matrix into simpler components
  • Used for solving systems, computing determinants, and preconditioning

Further Reading and Resources

• "Linear Algebra and Its Applications" by Gilbert Strang
  • Comprehensive textbook covering linear algebra concepts and applications
• "Matrix Analysis" by Roger A. Horn and Charles R. Johnson
  • In-depth treatment of matrix theory and matrix inequalities
• "Numerical Linear Algebra" by Lloyd N. Trefethen and David Bau III
  • Focuses on numerical methods and algorithms for linear algebra problems
• "Linear Algebra Done Right" by Sheldon Axler
  • Emphasizes the conceptual aspects of linear algebra with a more abstract approach
• MIT OpenCourseWare: Linear Algebra (18.06)
  • Lecture videos, notes, and assignments from the popular MIT course taught by Gilbert Strang
• Khan Academy: Linear Algebra
  • Interactive tutorials and practice problems covering linear algebra concepts
• MATLAB and Python (NumPy/SciPy) documentation
  • Software tools for implementing and visualizing linear algebra concepts and algorithms
• "Coding the Matrix: Linear Algebra through Applications to Computer Science" by Philip N. Klein
  • Introduces linear algebra concepts through programming applications in computer science