If S is an n \times n symmetric matrix and if x is the column of coordinates of a point p in an n-dimensional vector space relative to a given basis, then the matrix product
| x' S x |
is a 1 \times 1 matrix whose sole entry Q_{S}(p) is a scalar function of the point p that is a polynomial of degree 2 in the coordinates x_{1}, x_{2}, ..., x_{n} of p. The function Q_{S} is called a quadratic form, and S is the matrix of the quadratic form relative to the given coordinate system.
If with a change of basis each point p that is represented in a given basis by x is represented relative to another basis by y where x = A y for a given invertible matrix A, what is the matrix relative to the second basis of the quadratic form that has matrix S relative to the given basis?
Let f(x_{1}, x_{2}) = 2 x_{1} x_{2}.
For what 2 \times 2 symmetric matrix S is Q_{S} = f ?
Find a basis of R^{2} consisting of mutually perpendicular unit vectors relative to which the matrix of f is a diagonal matrix.
Let S be the 3 \times 3 symmetric matrix
| . |
Find a diagonal matrix that represents Q_{S} relative to some basis of R^{3} consisting of mutually perpendicular unit vectors.
What is the largest value achieved by Q_{S} on the unit sphere x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1 ?