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.

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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.

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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.

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10. Quadratic forms and quadratic surfaces

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.

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