Homework 0 - Solution

  1. Complex Numbers

    1. \(z^* = a - ib\)

    2. \((a + c) + i(b + d)\)

    3. \((ac - bd) + i(ad + bc)\)

    4. \( |z| = \sqrt{Re(z)^2 + Im(z)^2}\)
      In other words, if \(z = a + ib\), then \(|z| = \sqrt{a^2 + b^2}\)

    5. \(cos\ \theta + isin\ \theta\)

    6. \(|e^{i\theta}| = \sqrt{cos^2\ \theta + sin^2\ \theta} = 1\)

    7. \( i^2 = -1\\i^3 = -i\\i^4 = 1\)
    8. \(zz^* = (a + ib)(a - ib) = a^2 + b^2 = |z|^2\)

    9. \( e^{i\frac{\pi}{2}} = i\\e^{i\pi} = -1\\e^{i\frac{3\pi}{2}} = -i\\e^{i2\pi} = 1\)
    10. \(e^{\frac{2k\pi i}{n}}, k = 0,1,2,...n-1\)

  2. Linear Algebra

    1. \( \left(\begin{matrix}1.2\\3.5\\-0.05\\\end{matrix}\right) \) or \( \left[\begin{matrix}1.2\\3.5\\-0.05\\\end{matrix}\right]\)

    2. \( 1*4 + 2*5 + 3*6 = 32\)
      Note that the dot product of \(a\) and \(b\) can be written as \(a^\top b\).

    3. \( \mathbf{x}^\top = \left[\begin{matrix}1&2&3\end{matrix}\right]\)

    4. \( \mathbf{x}^\dagger = \left[\begin{matrix}a^*&b^*&c^*\end{matrix}\right]\)

    5. \( |a|^2 + |b|^2 + |c|^2\)

    6. \( \left[\begin{matrix}ax&ay&az\\bx&by&bz\\cx&cy&cz\end{matrix}\right]\)

      This is known as an outer product. The outer product of \(a\) and \(b\) is \(ab^\top\).

    7. \( \left[\begin{matrix}ax + by\\cx + dy\end{matrix}\right]\)

    8. \( \left[\begin{matrix}ax_0 + by_0 + cz_0&&ax_1 + by_1 + cz_1&ax_2 + by_2 + cy_2\end{matrix}\right] \)

    9. \( \left[\begin{matrix}ar + bu + cx&as + bv + cy&at + bw + cz\\dr + eu + fx&ds + ev + fy&dt + ew + fz\\hr + ju + kx&hs + jv + ky&ht + jw + kz\end{matrix}\right]\)

    10. \( I = \left[\begin{matrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}\right]\)

    11. \( A^{-1}A = AA^{-1} = I \)

    12. No, matrix multiplication is not commutative.

    13. \( \lambda \) is an eigenvalue of \(A\).

    14. \(\mathbf x\) is an eigenvector of \(A\).

    15. \( A^\top = \left[\begin{matrix}a&c\\b&d\end{matrix}\right]\)

    16. \( A^\dagger = \left[\begin{matrix}a^*&c^*\\b^*&d^*\end{matrix}\right]\ \)

    17. \( \left[\begin{matrix}0&1&0&0\\0&0&0&1\\0&0&1&0\\1&0&0&0\\\end{matrix}\right] \left[\begin{matrix}a\\b\\c\\d\\\end{matrix}\right] = \left[\begin{matrix}b\\d\\c\\a\end{matrix}\right]\)

    18. \( X^\dagger = \left[\begin{matrix}0&1\\1&0\\\end{matrix}\right],\ X^\dagger X = \left[\begin{matrix}0&1\\1&0\\\end{matrix}\right]\left[\begin{matrix}0&1\\1&0\\\end{matrix}\right] = \left[\begin{matrix}1&0\\0&1\\\end{matrix}\right]\)

      \( Y^\dagger = \left[\begin{matrix}0&-i\\i&0\\\end{matrix}\right],\ Y^\dagger Y = \left[\begin{matrix}0&-i\\i&0\\\end{matrix}\right]\left[\begin{matrix}0&-i\\i&0\\\end{matrix}\right] = \left[\begin{matrix}1&0\\0&1\\\end{matrix}\right]\)

      \( Z^\dagger = \left[\begin{matrix}1&0\\0&-1\\\end{matrix}\right],\ Z^\dagger Z = \left[\begin{matrix}1&0\\0&-1\\\end{matrix}\right]\left[\begin{matrix}1&0\\0&-1\\\end{matrix}\right] = \left[\begin{matrix}1&0\\0&1\\\end{matrix}\right] \)

      \( H^\dagger = \frac{1}{\sqrt{2}}\left[\begin{matrix}1&1\\1&-1\\\end{matrix}\right],\ H^\dagger H = \frac{1}{2}\left[\begin{matrix}1&1\\1&-1\\\end{matrix}\right]\left[\begin{matrix}1&1\\1&-1\\\end{matrix}\right] = \frac{1}{2}\left[\begin{matrix}2&0\\0&2\\\end{matrix}\right] = \left[\begin{matrix}1&0\\0&1\\\end{matrix}\right] \)


    19. The trace of a square matrix is the sum of the elements on the main diagonal: \( a_{11} + a_{22} + a_{33} \).