Liner Algebra Projects
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Matrix Addition and Subtraction Description: Basic operations to add or subtract two matrices of the same dimensions element-wise. Applications: Image processing, network theory, and data analysis.
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Matrix Multiplication Description: The product of two matrices results in another matrix, combining the rows of the first matrix with the columns of the second. Applications: Graphics transformations, neural networks, and computer vision.
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Scalar Multiplication Description: Multiplying a matrix by a scalar value, scaling each element of the matrix. Applications: Signal processing and scaling operations in graphics.
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Determinant of a Matrix Description: A scalar value that can be computed from a square matrix, indicating whether the matrix is invertible. Applications: Calculating volumes, solving systems of equations, and cryptography.
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Matrix Transpose Description: Flipping a matrix over its diagonal, switching its rows and columns. Applications: Data preprocessing, covariance matrices, and machine learning.
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Inverse of a Matrix Description: Finding a matrix that, when multiplied with the original matrix, results in the identity matrix. Applications: Solving linear systems, cryptography, and control systems.
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Solving Systems of Linear Equations (Gaussian Elimination) Description: An algorithm to solve systems of linear equations by reducing the matrix to row echelon form. Applications: Engineering, physics, and computer graphics.
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LU Decomposition Description: Decomposing a matrix into the product of a lower triangular matrix and an upper triangular matrix. Applications: Efficiently solving systems of equations and matrix inversion.
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QR Decomposition Description: Decomposing a matrix into an orthogonal matrix Q and an upper triangular matrix R. Applications: Solving linear systems, least squares problems, and eigenvalue computations.
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Eigenvalues and Eigenvectors Description: Finding the eigenvalues (scalars) and eigenvectors (directions) of a matrix. Applications: Principal Component Analysis (PCA), quantum mechanics, and stability analysis.
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Singular Value Decomposition (SVD) Description: Decomposing a matrix into three matrices: U, Σ, and V^T Applications: Image compression, noise reduction, and dimensionality reduction.
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Cholesky Decomposition Description: Decomposing a positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. Applications: Solving linear systems, numerical simulations, and optimization.
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Rank of a Matrix Description: The maximum number of linearly independent rows or columns in a matrix. Applications: Data compression, information theory, and machine learning.
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Moore-Penrose Pseudoinverse Description: Generalizing the concept of matrix inversion for non-square matrices. Applications: Linear regression, least squares problems, and machine learning.
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Norms of Vectors and Matrices Description: Measures of the length or size of vectors and matrices (e.g., Euclidean norm, Frobenius norm). Applications: Optimization, machine learning, and signal processing.
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Diagonalization of a Matrix Description: Converting a matrix into a diagonal form using its eigenvectors. Applications: Differential equations, dynamical systems, and Markov processes.
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Power Method for Eigenvalues Description: An iterative algorithm to find the largest eigenvalue of a matrix. Applications: PageRank algorithm, spectral graph theory, and principal component analysis.
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Matrix Exponentiation Description: Computing the exponential of a matrix, often used in solving systems of differential equations. Applications: Network analysis, control theory, and economics.
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Least Squares Regression (Linear Regression) Description: Finding the best-fitting line (or hyperplane) to minimize the sum of the squared residuals. Applications: Data science, machine learning, and predictive modeling.
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Householder Transformation Description: A reflection transformation used to zero out sub-diagonal elements of a matrix. Applications: QR decomposition and solving least squares problems.