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In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties.〔 〕 Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices),〔 〕 and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory). ==Matrix spaces== The set of all ''m''×''n'' matrices over a number field ''F'' denoted in this article ''M''''mn''(''F'') form a vector space. Examples of ''F'' include the set of integers ℤ, the real numbers ℝ, and set of complex numbers ℂ. The spaces ''M''''mn''(''F'') and ''M''''pq''(''F'') are different spaces if ''m'' and ''p'' are unequal, and if ''n'' and ''q'' are unequal; for instance ''M''32(''F'') ≠ ''M''23(''F''). Two ''m''×''n'' matrices A and B in ''M''''mn''(''F'') can be added together to form another matrix in the space ''M''''mn''(''F''): : and multiplied by a ''α'' in ''F'', to obtain another matrix in ''M''''mn''(''F''): : Combining these two properties, a linear combination of matrices A and B are in ''M''''mn''(''F'') is another matrix in ''M''''mn''(''F''): : where ''α'' and ''β'' are numbers in ''F''. Any matrix can be expressed as a linear combination of basis matrices, which play the role of the basis vectors for the matrix space. For example, for the set of 2×2 matrices over the field of real numbers, ''M''22(ℝ), one legitimate basis set of matrices is: : because any 2×2 matrix can be expressed as: : where ''a'', ''b'', ''c'',''d'' are all real numbers. This idea applies to other fields and matrices of higher dimensions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「matrix analysis」の詳細全文を読む スポンサード リンク
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