[QF] 01-Linear algebra

List of topics for quantitative finance with concise key points and useful resources. For Quant interview preparation and quantitative finance introduction, or could be cheatsheets. Motivated by Ran Ding.

resources

• GRA6035 Mathematics, M.Sc., BI
• DRE7017 Mathematics, Ph.D., BI
• Mathwords: Algebra to Calculus

key points

• Matrix basic
  • a m*n matrix has m rows and n columns
    • entry/element in matrix A: $a_{i,j}$ is in row i, column j
  • determinant of square, |A|: do row operations until the matrix is upper triangular
    • row operations: exchange two rows will change the sign; and others
    • determinant of a upper triangular matrix is the product of the diagonal entries/elements
  • identify matrix $E$ (diagonal elements 1, others 0) and diagonal matrix (diagonal elements not 0, others 0)
  • Minor $M_{i,j}$: matrix except $a_{i,j}$
    • Cofactor $A_{i,j}$ = $(-1)^{i+j}|M_{i,j}|$
    • Cofactor matrix $A^*$, cofactors term-by-term
• Matrix operation
  • matrix A m*s, matrix B s*n, C=AB, matrix C m*n
    • $c_{ij}$ = $\sum^s_{k=1}a_{ik}b_{kj}$
  • transpose $A^T$: $a^T_{j,i}$ = $a_{i,j}$
    • $(AB)^T$ = $B^TA^T$
  • inverse matrix $A^{-1}$: $A^{-1}A=E$
    • $(AB)^{-1}$ = $B^{-1}A^{-1}$
    • $A^{-1}$ = $\cfrac{A^{*T}}{|A|}$
  • A could be transformed into B, $A\sim B$
    • $A\sim B \Leftrightarrow PAQ=B$
  • echelon form, reduced echelon form
    • pivot: first non-zero element in echelon form
  • rank of matrix
    • the maximum number of linearly independent row vectors in the matrix
    • full rank $\Leftrightarrow$ invertible
• Types of matrices
  • symmetric matrix: $A^T$ = $A$
  • orthogonal matrix $\Leftrightarrow$ $A^TA=AA^T=E$ $\Leftrightarrow$ $A^{-1}=A^T$
• Eigenvalue and eigenvectors
  • $Ax=\lambda x$
  • solve $|A-\lambda E|=0$ get eigenvalues
  • then for each eigenvalue solve $Ax=\lambda x$ get eigenvectors
  • $trace(A)=\sum^n_{i=1}\lambda_i=\sum^n_{i=1}a_{i,i}$
  • $\prod^n_{i=1}=|A|$
• Hessian matrix
  • $$
    H(f)=\begin{pmatrix}
    f_{1,1}^{''} & f_{1,2}^{''} & \cdots & f_{1,n}^{''} \
    f_{2,1}^{''} & f_{2,2}^{''} & \cdots & f_{2,n}^{''} \
    \vdots & \vdots & \ddots & \vdots \
    f_{n,1}^{''} & f_{n,2}^{''} & \cdots & f_{n,n}^{''}
    \end{pmatrix}
    $$
  • f is a function with n variables
• Matrix decomposition
  • see Linear Algebra and Matrix Decompositions