matrices have the fundamental property that the change-of-base formula is the same for a symmetric bilinear form and the endomorphism that is represented by the same symmetric matrix. Spectral theorem asserts that, given such a symmetric matrix, there is an orthogonal change of basis such that the resulting matrix (of both the bilinear form and the endomorphism) is a diagonal matrix with the eigenvalues of the initial matrix on the diagonal. It follows that, over the reals, if the matrix of an endomorphism is symmetric, then it is diagonalizable. See also Active and passive transformation Covariance and contravariance of vectors Integral transform, the continuous analogue of change of basis. Notes References Bibliography External links MIT Linear Algebra Lecture on Change of Basis, from MIT OpenCourseWare Khan Academy Lecture on Change of Basis, from Khan Academy Linear algebra Matrix theory