Theorem. For every symmetric linear map from a finite-dimensional inner product space to itself there is an orthonormal basis of the inner product space consisting of characteristic vectors of the symmetric linear map.
Corollary. A symmetric real n \times n matrix may be conjugated with an orthogonal matrix into a diagonal matrix.
Definition. A symmetric bilinear form on a vector space V is a real-valued function F of two variables from V that satisfies all but the positivity condition in the definition of inner product. (See the handout of April 12.) A symmetric bilinear form has a matrix with respect to a given basis just as an inner product does except that the matrix need not be positive-definite. (See the handout dated April 26.) The matrices of a symmetric bilinear form relative to two different bases are related in the same way as the matrices of an inner product relative to two different bases.
Corollary. For every symmetric bilinear form on a finite-dimensional inner product space there is an orthonormal basis of the inner product space in which the matrix of the symmetric bilinear form is diagonal. The symmetric bilinear form is itself an inner product if and only if each of the diagonal entries of its matrix, relative to such a basis, is positive.
Special Case. When the number B^{2} - 4 A C is non-zero, the equation A x^{2} + B x y + C y^{2} = K in two variables admits a ``rotation of the coordinate axes'' in the plane that converts it to the standard form
with k = 1 unless K = 0, in which case k = 0.
Correction. Modify the last phrase of the statement of the theorem on the handout of April 30 to read: ``i.e., each vector in the given basis generates a 1-dimensional subspace of a characteristic subspace.''
Let A be the 2 \times 2 matrix
| 1 | 2 |
| 2 | -1 |
Find an orthonormal basis of R^{2} relative to which the matrix of the linear transformation x -> A x is diagonal.
What type of conic section is represented by the planar equation x^{2} + 4 x y - y^{2} = 1 ?
Let f be the linear transformation that is defined by f(x) = M x where M is the 3 \times 3 matrix
| 0 | 1 | 2 |
| 1 | 0 | 1 |
| 2 | 1 | 0 |
Verify that -2 is a proper value of f.
Find the characteristic polynomial phi_{M} of M.
Verify that -2 is a root of phi_{M}.
Determine all proper values of f.
For each proper value c of f find the characteristic subspace of f for c.
Explain why the three characteristic subspaces of f must be orthogonal to each other.
To what diagonal matrix is M similar?
Find an orthogonal matrix U for which U^{-1} M U is diagonal.