Matrix analysis

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).

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 M₃₂(F) ≠ M₂₃(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):

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