Sylvester's Law of Inertia

Sylvester’s Law of Inertia states that when you change a matrix into a simpler form (like diagonalizing it), the number of positive, negative, and zero values on its diagonal—the signs of its eigenvalues—remains unchanged. These counts are called the matrix's "inertia." Essentially, no matter what invertible transformations you apply, the fundamental nature of the matrix’s positive and negative properties stays the same. This concept helps us understand the inherent characteristics of quadratic forms and symmetric matrices, providing insight into stability and types of solutions in various mathematical and physical problems.