Eigenvalues and Eigenvectors
Definition
Let’s assume that a square matrix $A_{n \times n} = (a_{ij})$ is given.
- The determinant $M_{ij}$ of the matrix obtained by removing the $i$-th row and the $j$-th row from $A$ is called the minor.
- $C_{ij} := (-1)^{i + j} M_{ij}$ is referred to as the cofactor.
- The matrix of cofactors $C = \left( C_{ij} \right)$ and its transpose $C^{T}$ is called the classical adjugate matrix, represented by $\text{adj} (A)$.
Explanation
The most widely known result where the cofactor is used is undoubtedly the Laplace expansion. As can be inferred from the Adjugate’s prefix Adj-, it seems that in the past it was simply called the adjoint matrix, but in modern times, since the conjugate transpose matrix is often called the adjoint matrix, it seems to be called the classical adjugate matrix to distinguish it from that.
Properties
The following holds for the identity matrix $I$ and the determinant $\det$: $$ A \text{adj} (A) = \det (A) I $$ Especially, if $A$ is an invertible matrix, the classical adjugate matrix can be represented as follows: $$ \text{adj} (A) = \det (A) A^{-1} $$