Matrix Factorisations

Motivation: why this matters in quant finance

Factorisations are how quants solve linear systems without pretending every matrix should be inverted directly. Cholesky turns covariance matrices into simulation engines, QR stabilises least squares, and SVD exposes low-rank risk structure.

The informal idea

A factorisation rewrites one matrix as a product of simpler matrices. The product stores the same linear map but makes a specific task easier: solving, simulating, compressing, or diagnosing conditioning.

Formal definitions

Common decompositions include $A=LU$, $A=QR$, $\Sigma=LL^\top$ for positive definite covariance matrices, and $A=U\Sigma V^\top$ for singular value decompositions.

Key properties

Structure replaces inversion

Solving triangular systems is cheaper and stabler than forming $A^{-1}$.

Cholesky is the covariance workhorse

If $\Sigma=LL^\top$, then $L\mathbf{Z}$ has covariance $\Sigma$ when $\mathbf{Z}$ has identity covariance.

SVD separates signal scale by direction

Large singular values identify directions where the matrix acts strongly.

Worked example

Let $\Sigma=\begin{bmatrix}1&0.6\\0.6&1\end{bmatrix}$. A Cholesky factor is $L=\begin{bmatrix}1&0\\0.6&0.8\end{bmatrix}$ because $LL^\top=\Sigma$.

Common confusions and pitfalls

"Factorisation is just notation." It changes the numerical problem.

"Matrix inverse is the default tool." In computation, solve systems instead.

"Every factorisation exists for every matrix." Each has hypotheses or pivoting caveats.

Where this goes next

• Positive Definite Matrices: gives the condition behind Cholesky covariance factors.
• Eigenvalues and Eigenvectors: explains spectral decompositions.
• Covariance Matrices: applies factorisations to risk and simulation.

References

• Lang, S. (1986). Introduction to Linear Algebra (2nd ed.). Springer. Ch. II, Ch. V-VIII, and the eigenvalue chapter as relevant.