Linear algebra and matrices M3P9

Question Sheet 7

1. Prove the following statements:

(1) If A is a symmetric invertible matrix, then so is A^{−1}.

(2) If A is a real invertible matrix, then AA^{t} is symmetric and positive definite. (A matrix B is called positive definite if the corresponding bilinear form u^{t}.B.v is positive definite, that is, if u^{t}.B.u > 0 for any non-zero vector u.)

(3) If A is a real symmetric matrix, then there exists a real constant c such that A + c.Id is positive definite.

2. Let A be a real skew-symmetric matrix.

(1) Prove that Id+A is invertible. (Hint: use a property of the eigenvalues of skew-symmetric matrices from Sheet 6.)

(2) Deduce that (Id − A)(Id + A)^{−1} is an orthogonal matrix.

3. Prove that all eigenvalues of an orthogonal matrix with real coefficients are complex numbers with absolute value 1.

4. Prove that a hyperboloid with 2 sheets cannot contain an interval of a straight line.

5. Can a hyperbolic paraboloid contain a pair of (real) parallel lines?