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Today, it is an indispensable part of the discipline’s subject matter. Matrix/linear algebra benefits began accruing in the theoretical and applied statistics literature in the early 1940 s, when Fisher, and then Wilks, used the word matrix for the first time in their publications. The unfolding of the statistics field over recent centuries discloses a transformation of the role of matrix algebra from a mostly notational tool for conveniently expressing statistical problems, to an essential component in the conceptualization, derivation, comprehension, and utilization of more mature, complex, and complicated analytical devices characterizing the modern-day statistical sciences. These are the five prospectively reviewed discipline of statistics subjects capable of informing, inspiring, or otherwise furnishing insight to the far more general world of linear algebra. A fifth, and the final idea this paper treats, is random matrices/vectors within the context of simulation, particularly for correlated data. A fourth is multivariable calculus linkages, especially arcane matrix/vector operators as well as the Jacobian term associated with variable transformations. A third is special matrix operations, such as Kronecker sums and products. Another prominent one is the mathematical matrix theory eigenfunction abstraction. Not surprisingly, regression, the workhorse of statistics, provides one tool for such historically based recompence.

Nevertheless, statistics offers great potential for making this interface a bidirectional exchange point, the theme of this review paper. Contributions mostly have been unidirectional, from matrix/linear algebra to statistics. Consequently, focused parts of matrix algebra are topics of several statistics books and journal articles. Matrix/linear algebra continues bestowing benefits on theoretical and applied statistics, a practice it began decades ago (re Fisher used the word matrix in a 1941 publication), through a myriad of contributions, from recognition of a suite of matrix properties relevant to statistical concepts, to matrix specifications of linear and nonlinear techniques.