MATRIX DECOMPOSITION Meaning and
Definition
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Matrix decomposition refers to the process of breaking down a given matrix into multiple constituent parts or factors with specific properties. It involves expressing a matrix as a product of simpler matrices or vectors, which aids in simplifying calculations, solving mathematical problems, and understanding the underlying structure or properties of the matrix.
There are several types of matrix decompositions, each with its own characteristics and applications. One commonly used decomposition is the LU (Lower-Upper) decomposition, where a matrix is factored into a lower triangular matrix and an upper triangular matrix. LU decomposition is useful for solving systems of linear equations or calculating matrix inverse.
Another widely employed decomposition is the Singular Value Decomposition (SVD), which represents a matrix as a product of three matrices: a unitary matrix, a diagonal matrix with singular values, and the conjugate transpose of another unitary matrix. SVD is used for data compression, approximation, and dimensionality reduction.
Eigenvalue decomposition or eigendecomposition involves expressing a matrix as a product of eigenvectors and eigenvalues. It helps in understanding the properties of the matrix, diagonalization, and determining stability in dynamical systems.
Other notable matrix decompositions include QR decomposition, Cholesky decomposition, and Schur decomposition. Each decomposition method has its own mathematical properties, uses, and limitations. Matrix decomposition plays a crucial role in various fields such as linear algebra, numerical analysis, computer graphics, signal processing, machine learning, and scientific computing.
Etymology of MATRIX DECOMPOSITION
The word "matrix" comes from the Latin word "matrix", which means "womb" or "breeding place". It was later adopted to refer to a rectangular array of numbers or symbols.
The word "decomposition" comes from the Latin word "decompositio", which means "a breaking down" or "separation". It is derived from the verb "decomponere", which means "to break down" or "to undo".
When combined, "matrix decomposition" refers to the process of breaking down or decomposing a matrix into simpler and more manageable parts or components.
