User:IssaRice/Linear algebra/Properties preserved under various operations to a matrix
properties to discuss: characteristic polynomial, image, kernel, rank, nullity, determinant, invertibility, diagonalizability, eigenvectors, eigenvalues, normality
| Operation | Preserved properties | Properties not necessarily perserved |
|---|---|---|
| Elementary operation: swapping rows | determinant | |
| Elementary operation: shear | determinant | |
| Elementary operation: scaling one row | determinant | |
| Transpose | ||
| Inverse | ||
| Formation of adjoint (conjugate transpose) | ||
| Multiplication by an invertible matrix | ||
| Multiplication by an elementary matrix | ||
| Change of basis ( for some invertible ) | characteristic polynomial |
inspired by questions 5 and 6 at http://scherk.pbworks.com/w/page/14864231/Quiz%3A%20Linear%20algebra