form, there is an orthogonal diagonalization, that is an orthogonal change of variables that puts the quadratic form in a "diagonal form" where the associated symmetric matrix is diagonal. Moreover, the coefficients are determined uniquely up to a permutation. If the change of variables is given by an invertible matrix, that is not necessarily orthogonal, one can suppose that all coefficients are 0, 1, or −1. Sylvester's law of inertia states that the numbers of each 1 and −1 are invariants of the quadratic form, in the sense that any other diagonalization will contain the same number of each. The signature of the quadratic form is the triple , where n0 is the number of 0s and n± is the number of ±1s. Sylvester's law of inertia shows that this is a well-defined quantity attached to the quadratic form. The case when all λi have the same sign is especially important: in this case the quadratic form is called positive definite (all 1) or negative definite (all −1). If none of the terms are 0, then the form is called ; this includes positive definite, negative definite, and indefinite (a mix of 1 and −1); equivalently, a nondegenerate quadratic form is one whose associated symmetric form is a nondegenerate bilinear form. A real vector space with an indefinite nondegenerate quadratic form of index (denoting p 1s and q −1s) is often denoted as Rp,q particularly in the physical theory of spacetime. The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K/(K×)2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only “positive, zero, or negative”. Zero corresponds to degenerate, while for a non-degenerate form it is the parity of the number of negative coefficients, These results are reformulated in a different way below. Let q be a quadratic form defined on an n-dimensional real vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric matrix such that where x is the column vector of coordinates of v in the chosen basis. Under a change of basis, the column x is multiplied on the left by an invertible matrix S, and the symmetric square matrix A is transformed into another symmetric square matrix B of the