^^ [[Schutz - 1 - Some Basic Mathematics|1. Some Basic Mathematics]] << [[Schutz - 1.5 - Linear Algebra]] | # The Algebra of Square Matrices > **Determinant** > The determinant of a square matrix is defined by: > $\det(A) = \left(\sum_{j=1}^n a_{ij} a^{ij}\right) \quad \text{for any fixed} \, i $ ^9a7405 > **Transpose** > The transpose of a matrix is defined by: > $(A^T)_{ij} = A_{ji}$ > **Inverse** > The *inverse* of a square matrix is defined by: > $A^{-1} A = A A^{-1} = I$ > When the inverse exists it is unique. ^8fdccb > **Trace** > The *trace* of a square matrix is defined by > $\text{tr}(A) = \sum_{i=1}^n a_{ii}$ > **Non-singular Matrix** > A matrix $A$ is said to be *non-singular* when its [[Schutz - 1.6 - The Algebra of Square Matrices#^8fdccb|inverse]] exists. This is equivalent to saying that the [[Schutz - 1.6 - The Algebra of Square Matrices#^9a7405|determinant]] is non-zero. ^9ea58f > **Similarity Transformation** > A *similarity transformation* of a matrix $A$ by a [[Schutz - 1.6 - The Algebra of Square Matrices#^9ea58f|non-singular]] matrix $B$ is a map $A \mapsto B^{-1} A B$. ## Properties of Operations on Square Matrices $ \begin{align} (AB)^T &= B^T A^T, \\ (AB)^{-1} &= B^{-1} A^{-1}, \\ \det(AB) &= \det(A) \det(B), \\ \det(B^{-1} A B) &= \det(A), \\ \text{tr}(B^{-1} A B) &= \text{tr}(A),\\ \det(A^T) &= \det(A),\\ {\text{eigenvalues of} \, A^T} &= {\text{eigenvalues of} \, A} , \\ \det(A) &= \lambda_1 \lambda_2 \ldots \lambda_n \\ \text{tr}(A) &= \lambda_1 +\lambda_2 +\ldots +\lambda_n \end{align} $