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 \(A.\)