The sum and product of diagonal matrices are again diagonal, and the diagonal matrices form a subring of the ring of square matrices: indeed for n×n matrices over a ring R this ring is isomorphic to the product ring Rn.
The diagonal entries are the eigenvalues of a diagonal matrix.
The determinant of a diagonal matrix is the product of the diagonal elements.
A diagonalizable matrix is a square matrix which is similar to a diagonal matrix: that is, A is diagonalizable if there exists an invertible matrix P such that P − 1AP is diagonal. The following conditions are equivalent:
- A is diagonalizable;
- The minimal polynomial of A has no repeated roots;
- A is n×n and has n linearly independent eigenvectors.
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