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Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series and analysis functions.
Mathematical analysis of data was a well-established process in science when Kepler began studying Tycho's data.
Analysis can be thought of as a continuation of calculus. It deals with topics such as measure, limits, and integration/differentiation, and spaces (such as metric spaces).
Issac Newton and Gottfried Leibniz are the co-discovers of this system of mathematical analysis.
Eigenvalues and eigenvectors are properties of a mathematical matrix.See related Wikipedia link for more details on what they are and some examples of how to use them for analysis.