is a change-of-basis matrix, then the matrix of the endomorphism on the "new" basis is As every invertible matrix can be used as a change-of-basis matrix, this implies that two matrices are similar if and only if they represent the same endomorphism on two different bases. Bilinear forms A bilinear form on a vector space V over a field is a function which is linear in both arguments. That is, is bilinear if the maps and are linear for every fixed The matrix of a bilinear form on a basis (the "old" basis in what follows) is the matrix whose entry of the th row and th column is . It follows that if and are the column vectors of the coordinates of two vectors and , one has where denotes the transpose of the matrix . If is a change of basis matrix, then a straightforward computation shows that the matrix of the bilinear form on the new basis is A symmetric bilinear form is a bilinear form such that for every and in . It follows that the matrix of on any basis is symmetric. This implies that the property of being a symmetric matrix must be kept by the above change-of-base formula. One can also check this by noting that the transpose of a matrix product is the product of the transposes computed in the reverse order. In particular, and the two members of this equation equal if the matrix is symmetric. If the characteristic of the ground field is not two, then for every symmetric bilinear form there is a basis for which the matrix is diagonal. Moreover, the resulting nonzero entries on the diagonal are defined up to the multiplication by a square. So, if the ground field is the field of the real numbers, these nonzero entries can be chosen to be either or . Sylvester's law of inertia is a theorem that asserts that the numbers of and of depends only on the bilinear form, and not of the change of basis. Symmetric bilinear forms over the reals are often encountered in geometry and physics, typically in the study of quadrics and of the inertia of a rigid body. In these cases, orthonormal bases are specially useful; this means that one generally prefer to restrict changes of basis to those that have an orthogonal change-of-base matrix, that is, a matrix such that Such