Exam 11: Additional Topics and Applications

Verify that the given matrix A is normal, then diagonalize A by finding a unitary matrix P and diagonal matrix D such that .

Find the maximum and minimum values of the quadratic form subject to the constraints and , where .

Show that the given matrix is positive definite, and then find the LDU-factorization.

Find a polynomial formula for the quadratic form with the matrix .

Find the maximum and minimum values of the quadratic form subject to the constraint .

If B is any matrix, then the formula defines a positive semidefinite quadratic form on .

Find the matrix A of the quadratic form .

Find for the matrix .

Determine if the matrix A is positive definite.

If m is the minimum value of a quadratic form subject to the constraint , then cm is the minimum value of subject to the constraint .

If A is positive definite, every entry along the diagonal of A is positive.

If the quadratic form is positive definite, then A is invertible.

If A and B are matrices that are both normal, and , then both and AB are normal.

If A has an LDU-factorization, then A is positive definite.

Evaluate .

Show that the given matrix is positive definite, and then find the LDU-factorization of the matrix.

A symmetric matrix is positive definite if and only if is negative definite.

Evaluate for the matrices .

Find for the matrix .

Determine if is in .