chapter  9
58 Pages

Linear Groups

WithGarrett Birkhoff, Saunders Mac Lane

The coordinates of a vector ? in a space V depend upon the choice of a basis in V; hence any change in the basis will cause a change in the coordinates of ?. The algebra of matrices applies especially smoothly to diagonal matrices: to add or multiply any two diagonal matrices, one simply adds corresponding diagonal entries. For this and other reasons, it is important to know which matrices are similar to diagonal matrices. The answer to these questions involves the notions of characteristic vector and characteristic root—also called eigenvector and eigenvalue. Thus each eigenvector ? of T determines an eigenvalue c, and each eigenvalue belongs to at least one eigenvector. Since similar matrices correspond to the same linear transformation under different choice of bases, similar matrices have the same eigenvalues. All nonsingular linear transformations of an n-dimensional vector space Fn form a group because the products and inverses of such transformations are again linear and nonsingular.