Vectors and matrices: problem set 4 (intersection of planes and diagonalization)

5. Conditions for solutions

Find the value of $\alpha$ such that the three planes: \begin{align*} x + \alpha y &= 1 \\ x - y + 3z &= -1 \\ 2x - 2y + \alpha z &= -2, \end{align*}

have exactly one solution, no solutions, or an infinite number of solutions Give the form of the solutions for the last case.

Spoiler

6. Conditions for solutions

Find the value of $\alpha$ such that the three planes \begin{align*} 3x + 2y - z &= 10 \\ 5x - y - 4z &= 17 \\ x + 5y + \alpha z &= \beta, \end{align*}

do not intersect in a single point. Show that for this value of $\alpha$ there is no point common to all these planes unless $\beta = 3$. For the case of the three planes having a common line of intersection, find its equation in cartesian form.

Spoiler

9. Diagonalization of Hermitian matrices

Find the normalized eigenvectors of the Hermitian matrix $H = \begin{pmatrix} 10 & 3i \\ -3i & 2 \end{pmatrix}$

and construct a unitary matrix $U$ such that $U^\dagger H U = \Lambda$ where $\Lambda$ is a real diagonal matrix.

Spoiler

Show that the quadratic surface $5x^2 + 11 y^2 + 6z^2 - 10yz + 2xz - 10xy = 4,$
is an ellipsoid with semi-axes of lengths $2, 1$ and $0.5$. Find the direction of its longest axis.