Linear Algebra

An $m \times n$ matrix is a grid of numbers with $m$ rows and $n$ columns that can be regarded as function (linear transformation) from $\mathbf{R}^n$ to $\mathbf{R}^m$ via matrix multiplication. For a matrix $A$ we usually denote the $(i,j)$-entry of $A$ as $a_{ij}$ and write $A = (a_{ij})$ to express $A$ in terms of its entries. The $n \times n$ identity matrix, denoted $\mathrm{I}_n,$ has ones along its diagonal and zeros elsewhere, and corresponds to the identity function.

Given a matrix $A$ its transpose $A^t$ is the result of reflecting the entries of $A$ over its diagonal — the $(i,j)$-entry of $A^t$ is $a_{ji}.$ For a matrix $A$ with complex entries, its conjugate $\overline{A}$ is the matrix whose entries are the complex conjugates of those of $A.$ A matrix $A$ is symmetric if $A^t = A,$ skew-symmetric if $A^t = -A,$ and Hermitian if $A^t = \overline{A}.$ The inverse of a square matrix $A,$ denoted $A^{-1},$ is defined by the property that $AA^{-1} = A^{-1}A = \mathrm{I}_n.$ The trace of a square matrix $A,$ denoted $\operatorname{tr}(A),$ is the sum of the diagonal entries of $A.$ A fun property of the trace: for matrices $A$ and $B$ and $C,$ $\operatorname{tr}(ABC) = \operatorname{tr}(CAB) = \operatorname{tr}(BCA).$ The determinant of a matrix $A,$ denoted either $\operatorname{det}(A)$ or $\bigl|a_{ij}\bigr|,$ is equal to $$\sum_\sigma \biggl(\operatorname{sign}(\sigma)\prod_{i=1}^n a_{i\,\sigma(i)}\biggr)$$ where the sum is taken over all permutations $\sigma$ of $\{1, 2, \dotsc, n\}.$ Intuitively though the determinant is just how much the volume of space scales under the transformation $A;$ a negative determinant indicates a reflection of space.

Regarding $A$ as a linear transformation, for any vector $\bm{v}$ for which there exists a constant $\lambda$ such that $A(\bm{v}) = \lambda\bm{v}$ is called an eigenvector of $A,$ and $\lambda$ its corresponding eigenvalue. The trace of a matrix will be the sum of eigenvalues. The eigenvalues can also be defined as the roots of $\operatorname{det}\bigl(\lambda \mathrm{I}_n-A\bigr),$ the characteristic polynomial of $A.$