A multivector that is the exterior product of formula_61 linearly independent vectors is called a "blade", and is said to be of grade formula_61. A multivector that is the sum of blades of grade formula_61 is called a (homogeneous) multivector of grade formula_61. From the axioms, with closure, every multivector of the geometric algebra is a sum of blades. Consider a set of formula_61 linearly independent vectors formula_66 spanning an formula_61-dimensional subspace of the vector space. With these, we can define a real symmetric matrix (in the same way as a Gramian matrix)  By the spectral theorem, formula_69 can be diagonalized to diagonal matrix formula_70 by an orthogonal matrix formula_71 via  Define a new set of vectors formula_73, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:  Since orthogonal transformations preserve inner products, it follows that formula_75 and thus the formula_73 are perpendicular. In other words, the geometric product of two distinct vectors formula_77 is completely specified by their exterior product, or more generally  Therefore, every blade of grade formula_61 can be written as a geometric product of formula_61 vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a block matrix that is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace are normalized according to  then these normalized vectors must square to formula_82 or formula_83. By Sylvester's law of inertia, the total number of formula_82s and the total number of formula_83s along the diagonal matrix is invariant. By extension, the total number formula_86 of these vectors that square to formula_82 and the total number formula_88 that square to formula_83 is invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra formula_90. For example, formula_91 models formula_57-dimensional Euclidean space, formula_93 relativistic spacetime and formula_94 a conformal geometric algebra of a formula_57-dimensional space. The set of all possible products of formula_96 orthogonal basis vectors with indices in increasing order, including formula_18 as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem).