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Vectors, Matrices and What a Matrix Does

Linear algebra is the language of portfolios and regression, and research interviews test it directly: "compute the volatility of this two-asset portfolio", "why must a covariance matrix be positive semi-definite?", "can these three correlations exist together?". None of these needs heavy theory. Each needs a clear picture of what a vector and a matrix represent.

Portfolios are vectors

Hold nn assets with weights w=(w1,…,wn)w = (w_1, \ldots, w_n), and let their returns over a period be r=(r1,…,rn)r = (r_1, \ldots, r_n). The portfolio return is the dot product

w⊤r=∑i=1nwiriw^\top r = \sum_{i=1}^{n} w_i r_i

A matrix is a linear map: multiplying a vector by an m×nm \times n matrix turns nn numbers into mm numbers, and multiplying two matrices chains two maps. Dimensions must match, (m×n)(n×p)=m×p(m \times n)(n \times p) = m \times p, and the order matters: ABAB and BABA are generally different, when both exist at all.

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