Matrix Operations — Section 17: Linear Algebra

Matrices are linear transformations. The product $AB$ composes them: applying $B$ then $A$ . Composition is associative ( $A(BC) = (AB)C$ ) but in general not commutative ( $AB \ne BA$ ).

Key operations:

Transpose: $A^\top$ swaps rows and columns. $(AB)^\top = B^\top A^\top$ .
Inverse: $A^{-1}$ exists iff $A$ is square and full-rank (non-zero determinant). $(AB)^{-1} = B^{-1} A^{-1}$ .
Trace: $\mathrm{tr}(A) = \sum_i A_{ii}$ . Cyclic property $\mathrm{tr}(ABC) = \mathrm{tr}(BCA) = \mathrm{tr}(CAB)$ is endlessly useful.
Determinant: scaling factor of the linear map. $\det(AB) = \det(A) \det(B)$ .

In quant work, matrix algebra appears everywhere: $\hat{\beta} = (X^\top X)^{-1} X^\top y$ in regression, $\mathrm{Var}(w^\top X) = w^\top \Sigma w$ in portfolio variance, and Markov chain transitions $\pi P^n$ for state distributions.

Matrices are linear transformations. The product $AB$ composes them: applying $B$ then $A$ . Composition is associative ( $A(BC) = (AB)C$ ) but in general not commutative ( $AB \ne BA$ ).

Key operations:

Transpose: $A^\top$ swaps rows and columns. $(AB)^\top = B^\top A^\top$ .
Inverse: $A^{-1}$ exists iff $A$ is square and full-rank (non-zero determinant). $(AB)^{-1} = B^{-1} A^{-1}$ .
Trace: $\mathrm{tr}(A) = \sum_i A_{ii}$ . Cyclic property $\mathrm{tr}(ABC) = \mathrm{tr}(BCA) = \mathrm{tr}(CAB)$ is endlessly useful.
Determinant: scaling factor of the linear map. $\det(AB) = \det(A) \det(B)$ .