matrixanalyse

Matrix analysis, or matrixanalyse in German contexts, is a branch of linear algebra that studies properties and functions of matrices, primarily real or complex square matrices, through eigenvalues, eigenvectors, matrix functions, and inequalities. It combines algebraic structure with analytic techniques to understand how matrices act on spaces and how their spectral data governs stability and dynamics.

Key concepts include spectral decomposition for diagonalizable matrices, where a matrix is expressed via its eigenvalues and eigenvectors; the spectral theorem for Hermitian or real symmetric matrices; and unitary or orthogonal diagonalization for normal matrices. For many applications, positive definite or semi-definite matrices, with well-behaved square roots, play a central role in optimization and statistics.

Matrix factorizations such as the singular value decomposition, Schur decomposition, and, in some cases, Jordan form, provide canonical representations useful for computations and theory. Matrix functions extend scalar functions to matrices through series or functional calculus, enabling exponential, logarithm, and other transformations. Matrix calculus treats differentiation and optimization with respect to matrix variables.

Norms and inequalities, including operator, Frobenius, and spectral norms, measure size and conditioning. Perturbation theory studies how small changes in a matrix affect eigenvalues and singular values, with results like Weyl’s inequalities and perturbation bounds widely used in numerical analysis.

Applications span solving linear systems and differential equations, spectral clustering, Markov chains, statistics through covariance matrices, control theory, computer graphics, and quantum mechanics. The field intersects numerical linear algebra, functional analysis, and operator theory and remains active in both theoretical and computational directions.

Historical roots trace to the 19th century with Cayley, Sylvester, and Hermite, and the development of eigenvalue methods, matrix factorizations, and positivity criteria. Modern matrix analysis emphasizes rigorous inequalities, matrix functions, and the interplay between algebraic and analytic methods.