Vectors and matrices: problem set 3 (hermitian and orthogonal matrices)

6. Matrix proof

  1. If $A^2 = 0$, show that $(I - A)$ is invertible
  2. If $A^3 = 0$, show that $(I - A)$ is invertible
  3. If $A^n = 0$ for $n \in \mathbb{Z}^+$, show that $(I - A)$ is invertible
  4. If $A^2 + 2A + I = 0$, show that $A$ is invertible
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11. Hermitian proofs

Prove the following results:

  1. If $A$ is Hermitian and $U$ is unitary, then $U^{-1}AU$ is Hermitian.
  2. If $A$ is anti-Hermitian, then $iA$ is Hermitian.
  3. The product of two Hermitian matrices, $A$ and $B$ is Hermitian if and only if $A$ and $B$ commute.
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9. Trace and determinant

If $A$ is a non-singular $3\times 3$ matrix and $B = 2A^{-1}$, calculate $\text{Tr}(AB)$ and $|A| |B|$.

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10. Properties of determinants

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11. Proof for the rotation matrix

The points $\mathbf{v} = [x y]^T$ and $\mathbf{v}' = [x' y']^T$ are related by $\mathbf{v}' = A\mathbf{v}$, where \[ R = \begin{bmatrix} \cos\theta \sin\theta\\ -\sin\theta \cos \theta \end{bmatrix} \]

Show that $x^2 + y^2 = x'^2 + y'^2$ and find the inverse of $R$ (note that this shows that the original and transformed points lie on a circle around the origin).

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