Nice non-real unitary matrices

I will define a nice unitary matrix as one that is not real but where all entries have at most one or two digits beyond the decimal point.

In this document, we will look at

1. 2 × 2 non-real complex matrices
2. 3 × 3 non-real complex matrices
3. 4 × 4 non-real complex matrices

From these, you can construct nice matrices that have integer singular values.

Unitary 2 × 2 non-real complex matrices

The possible unitary matrices that have at most one or two digits beyond the decimal point fall into one of two categories:

1. matrices that have a pattern based on the values in one column, and
2. matrices that have each entry equalling $0.5$ in absolute value.

We will look at each of these here.

Case 1

First, given two complex numbers $\alpha$ and $\beta$, then the following matrix has orthogonal column vectors:

$\begin{pmatrix} \alpha & \beta^* \\ \beta & -\alpha^* \end{pmatrix}$

Also, you can swap rows or columns, and you can multiply any row or column by any complex number on the unit circle, including $1$, $0.8 + 0.6j$, $0.6 + 0.8j$, $j$, $-0.6 + 0.8j$, $-0.8 + 0.6j$, $-1$, $-0.8 - 0.6j$, $-0.6 - 0.8j$, $-j$, $0.6 - 0.8j$ and $0.8 - 0.6j$. Also, you can also conjugate both $\alpha$ and $\beta$.

You can create such $\alpha$ and $\beta$ by taking any four single digit numbers $m_1$, $m_2$, $m_3$ and $m_4$ such that $m_1^2 + m_2^2 + m_3^2 + m_4^2 = 100$, and letting $\alpha = \pm 0.m_1 \pm 0.m_2j$ and $\beta = \pm 0.m_3 \pm 0.m_4j$ will work. All such sorted quartets include:

                          [0, 0, 6, 8]
                          [1, 1, 7, 7]
                          [1, 3, 3, 9]
                          [1, 5, 5, 7]
                          [2, 4, 4, 8]
                          [5, 5, 5, 5]

For example,

 0.8+0.6j,  0
 0,        -0.8+0.6j

 0.1+0.1j,  0.7-0.7j
 0.7+0.7j, -0.1+0.1j

 0.1+0.3j,  0.9-0.3j
 0.3+0.9j, -0.3+0.1j

 0.1+0.5j,  0.7-0.5j
 0.5+0.7j, -0.5+0.1j

 0.2+0.4j,  0.4+0.8j
 0.4+0.8j, -0.2-0.4j

 0.5+0.5j,  0.5+0.5j
 0.5+0.5j, -0.5-0.5j

Unfortunately, there are no non-zero triplets that add up to $100$, and unfortunately, at least one digit is always repeated; but such is life.

For example, you could scramble the second-last quartet to come up with $\alpha = -0.4+0.2j$ and $\beta = -0.8-0.4j$, so the unitary matrix is

$\begin{pmatrix} -0.4 + 0.2j & -0.8+0.4j \\ -0.8-0.4j & 0.4+0.2j \end{pmatrix}$.

You can now take this matrix and 1) conjugate each entry, 2) multiply the second row by $j$ and 3) multiply the first column by $0.6 - 0.8j$:

$\begin{pmatrix} -0.4 + 0.2j & -0.8-0.4j \\ 0.88+0.16j & -0.2-0.4j \end{pmatrix}$.

For two digits beyond the decimal point, you can create such $\alpha$ and $\beta$ by taking any four two-digit numbers $m_1m_2$, $m_3m_4$, $m_5m_6$ and $m_7m_8$ such that $m_1m_2^2 + m_3m_4^2 + m_5m_6^2 + m_7m_8^2 = 10000$, and letting $\alpha = \pm 0.m_1m_2 + \pm 0.m_3m_4j$ and $\beta = \pm 0.m_5m_6 + \pm 0.m_7m_8j$ will work. All such sorted quartets include:

                        [00, 00, 28, 96]
                        [00, 36, 48, 80]
                        [00, 48, 60, 64]
                        [02, 02, 34, 94]
                        [02, 10, 50, 86]
                        [02, 14, 14, 98]
                        [02, 14, 70, 70]
                        [02, 22, 26, 94]
                        [02, 22, 46, 86]
                        [02, 26, 62, 74]
                        [02, 34, 38, 86]
                        [02, 34, 46, 82]
                        [02, 34, 58, 74]
                        [06, 06, 18, 98]
                        [06, 06, 62, 78]
                        [06, 10, 42, 90]
                        [06, 18, 54, 82]
                        [06, 42, 46, 78]
                        [06, 42, 62, 66]
                        [08, 08, 64, 76]
                        [08, 12, 24, 96]
                        [08, 16, 44, 88]
                        [08, 20, 56, 80]
                        [08, 24, 48, 84]
                        [08, 32, 56, 76]
                        [08, 40, 44, 80]
                        [08, 52, 56, 64]
                        [10, 10, 14, 98]
                        [10, 26, 50, 82]
                        [10, 30, 54, 78]
                        [10, 34, 62, 70]
                        [12, 24, 64, 72]
                        [14, 22, 22, 94]
                        [14, 22, 62, 74]
                        [14, 46, 62, 62]
                        [16, 16, 32, 92]
                        [16, 20, 40, 88]
                        [16, 32, 64, 68]
                        [18, 26, 30, 90]
                        [18, 26, 54, 78]
                        [18, 54, 54, 62]
                        [22, 26, 38, 86]
                        [22, 26, 46, 82]
                        [22, 26, 58, 74]
                        [22, 46, 50, 70]
                        [24, 44, 48, 72]
                        [26, 34, 38, 82]
                        [26, 46, 58, 62]
                        [28, 32, 64, 64]
                        [30, 30, 46, 78]
                        [30, 30, 62, 66]
                        [32, 40, 40, 76]
                        [32, 52, 56, 56]
                        [34, 34, 62, 62]
                        [34, 38, 50, 70]
                        [34, 46, 58, 58]
                        [34, 50, 50, 62]
                        [36, 48, 48, 64]
                        [40, 40, 52, 64]
                        [42, 42, 46, 66]

For example, you could scramble [14, 22, 62, 74] to come up with $\alpha = -0.74+0.22j$ and $\beta = -0.14-0.62j$, so the unitary matrix is

$\begin{pmatrix} -0.74 + 0.22j & -0.14-0.62j \\ -0.14+0.62j & 0.74+0.22j \end{pmatrix}$.

You'll notice that twenty-one of these contain two values that represent numbers that would have only one digit beyond the decimal point:

                        [00, 00, 28, 96]
                        [00, 36, 48, 80]
                        [00, 48, 60, 64]
                        [02, 10, 50, 86]
                        [02, 14, 70, 70]
                        [06, 10, 42, 90]
                        [08, 20, 56, 80]
                        [08, 40, 44, 80]
                        [10, 10, 14, 98]
                        [10, 26, 50, 82]
                        [10, 30, 54, 78]
                        [10, 34, 62, 70]
                        [16, 20, 40, 88]
                        [18, 26, 30, 90]
                        [22, 46, 50, 70]
                        [30, 30, 46, 78]
                        [30, 30, 62, 66]
                        [32, 40, 40, 76]
                        [34, 38, 50, 70]
                        [34, 50, 50, 62]
                        [40, 40, 52, 64]

For example, you could scramble [06, 10, 42, 90] to come up with $\alpha = -0.9+0.06j$ and $\beta = -0.42-0.1$, so the unitary matrix is

$\begin{pmatrix} -0.9 + 0.06j & -0.42-0.1 \\ -0.42+0.1 & 0.9+0.06j \end{pmatrix}$.

Case 2

Also note that $0.1^2 + 0.7^2 = 0.5^2 + 0.5^2 = 0.34^2 + 0.62^2 = 0.25$, so some matrices like

$\begin{pmatrix} 0.1 - 0.7j & -0.5-0.5j \\ -0.7-0.1j & 0.5-0.5j \end{pmatrix}$

will also be unitary or this matrix's conjugate transpose. This author must still work out all combinations of the relationships between the signs of the various real and imaginary components must be satisfied...

Unitary 3 × 3 non-real complex matrices

Given a triplet of points $\alpha = \alpha_1 + \alpha_2 j$, $\beta = \beta_1 + \beta_2 j$ and $\gamma = \frac{(2 \alpha_2 \beta_2 + \alpha_1 \beta_1)(\beta_1 - \alpha_1) + \alpha_2^2 \beta_1 - \alpha_1 \beta_2^2 + (2 \alpha_1 \beta_1 (\beta_2 - \alpha_2) + \alpha_2 \beta_2 (\beta_2 - \alpha_2) - \alpha_2*\beta_1^2 + \alpha_1^2*\beta_2)j}{\alpha\alpha^* - \beta\beta^*}$ where the sum of the absolute values of these three each squared equals 1, then the following is a unitary matrices:

$\begin{pmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{pmatrix}$

Of course, you can swap any two rows or columns, or multiply any row or column by any complex number on the unit circle, including $1$, $0.8 + 0.6j$, $0.6 + 0.8j$, $j$, $-0.6 + 0.8j$, $-0.8 + 0.6j$, $-1$, $-0.8 - 0.6j$, $-0.6 - 0.8j$, $-j$, $0.6 - 0.8j$ and $0.8 - 0.6j$.

Here is a list of such triplets that have at most one digit beyond the decimal point:

[0.2+0.8j, -0.4,  -0.4]
[0.2-0.8j, -0.4,  -0.4]
[0.8+0.2j, -0.4j, -0.4j]
[0.8-0.2j,  0.4j,  0.4j]


[0.2+0.4j, -0.6-0.2j, -0.6-0.2j]
[0.2-0.4j, -0.6+0.2j, -0.6+0.2j]
[0.2+0.4j,  0.2-0.6j,  0.2-0.6j]
[0.2-0.4j,  0.2+0.6j,  0.2+0.6j]

[0.4+0.2j, -0.6+0.2j, -0.6+0.2j]
[0.4+0.2j, -0.2-0.6j, -0.2-0.6j]
[0.4-0.2j, -0.2+0.6j, -0.2+0.6j]
[0.4-0.2j, -0.6-0.2j, -0.6-0.2j]


[0.8+0.2j, -0.4j, -0.4j]
[0.8-0.2j, 0.4j, 0.4j]

[0.2+0.8j, -0.4, -0.4]
[0.2-0.8j, -0.4, -0.4]


[0.2+0.4j,  0.2-0.6j,  0.2-0.6j]
[0.2+0.4j, -0.6-0.2j, -0.6-0.2j]
[0.2-0.4j,  0.2+0.6j,  0.2+0.6j]
[0.2-0.4j, -0.6+0.2j, -0.6+0.2j]

[0.4+0.2j, -0.2-0.6j, -0.2-0.6j]
[0.4+0.2j, -0.6+0.2j, -0.6+0.2j]
[0.4-0.2j, -0.2+0.6j, -0.2+0.6j]
[0.4-0.2j, -0.6-0.2j, -0.6-0.2j]

Here is a list of such triplets that have at most one digit beyond the decimal point:

[0.08+0.44j,  0.36-0.52j,  0.36-0.52j]
[0.08+0.44j, -0.52-0.36j, -0.52-0.36j]
[0.08-0.44j,  0.36+0.52j,  0.36+0.52j]
[0.08-0.44j, -0.52+0.36j, -0.52+0.36j]

[0.44+0.08j, -0.52+0.36j, -0.52+0.36j]
[0.44+0.08j, -0.36-0.52j, -0.36-0.52j]
[0.44-0.08j, -0.36+0.52j, -0.36+0.52j]
[0.44-0.08j, -0.52-0.36j, -0.52-0.36j]

[0.28+0.32j, -0.64j, -0.64j]
[0.28-0.32j,  0.64j,  0.64j]

[0.32+0.28j, -0.64, -0.64]
[0.32-0.28j, -0.64, -0.64]


[0.32-0.76j,  0.32+0.24j,  0.32+0.24j]
[0.32+0.76j,  0.32-0.24j,  0.32-0.24j]

[0.76+0.32j, -0.24+0.32j, -0.24+0.32j]
[0.76-0.32j, -0.24-0.32j, -0.24-0.32j]


[0.44-0.72j, -0.36-0.12j, -0.36-0.12j]
[0.44+0.72j, -0.36+0.12j, -0.36+0.12j]
[0.72-0.44j,  0.12+0.36j,  0.12+0.36j]
[0.72+0.44j,  0.12-0.36j,  0.12-0.36j]


[0.52-0.64j,  0.24+0.32j,  0.24+0.32j]
[0.52+0.64j,  0.24-0.32j,  0.24-0.32j]
[0.64-0.52j, -0.32-0.24j, -0.32-0.24j]
[0.64+0.52j, -0.32+0.24j, -0.32+0.24j]

It is also possible to find other examples, of which these are a small smattering:

 0.7+0.7j, -0.1j,     -0.1j
 0.1+0.1j,  0.7j,      0.7j
 0,         0.5-0.5j,  0.5+0.5j

 0.7+0.7j, -0.1j,     -0.1
 0.1+0.1j,  0.7j,      0.7
 0,         0.5-0.5j,  0.5-0.5j

 0.3+0.9j, -0.3j,     -0.1j
 0.1+0.3j,  0.9j,      0.3j
 0,         0.1-0.3j,  0.3+0.9j

 0.3+0.9j, -0.1-0.2j,  0.1-0.2j
 0.1+0.3j,  0.3+0.6j, -0.3+0.6j
 0,         0.1-0.7j,  0.1+0.7j

 0.3+0.9j, -0.1-0.2j,  0.1+0.2j
 0.1+0.3j,  0.3+0.6j, -0.3-0.6j
 0,         0.1+0.7j,  0.5+0.5j

 0.3+0.9j, -0.1-0.1j, -0.2-0.2j
 0.1+0.3j,  0.3+0.3j,  0.6+0.6j
 0,         0.4-0.8j,  0.4-0.2j

 0.3+0.9j, -0.1-0.2j,  0.1-0.2j
 0.1+0.3j,  0.3+0.6j, -0.3+0.6j
 0,         0.5-0.5j,  0.5+0.5j

 0.4+0.8j, -0.4j,     -0.2j
 0.2+0.4j,  0.8j,      0.4j
 0,         0.2-0.4j,  0.4+0.8j

 0.4+0.8j, -0.4j,     -0.2
 0.2+0.4j,  0.8j,      0.4
 0,         0.2-0.4j,  0.8-0.4j

 0.4+0.8j, -0.1-0.1j,  0.3-0.3j
 0.2+0.4j,  0.2+0.2j, -0.6+0.6j
 0,         0.3-0.9j,  0.1+0.3j

 0.4+0.8j, -0.1-0.3j,  0.1-0.3j
 0.2+0.4j,  0.2+0.6j, -0.2+0.6j
 0,         0.1-0.7j,  0.1+0.7j

 0.4+0.8j, -0.1-0.3j, -0.1-0.3j
 0.2+0.4j,  0.2+0.6j,  0.2+0.6j
 0,         0.1-0.7j,  0.5+0.5j

 0.4+0.8j, -0.1-0.3j,  0.1-0.3j
 0.2+0.4j,  0.2+0.6j, -0.2+0.6j
 0,         0.5-0.5j,  0.5+0.5j

 0.5+0.5j, -0.5j,      0.3-0.4j
 0.5+0.5j,  0.5j,     -0.3+0.4j
 0,         0.5-0.5j,  0.1+0.7j

 0.5+0.5j, -0.5j,      0.5
 0.5+0.5j,  0.5j,     -0.5
 0,         0.5+0.5j,  0.5+0.5j

 0.5+0.5j, -0.3-0.3j, -0.4+0.4j
 0.5+0.5j,  0.3+0.3j,  0.4-0.4j
 0,         0.8j,      0.6j

 0.5+0.5j, -0.1-0.6j, -0.3+0.2j
 0.5+0.5j,  0.1+0.6j,  0.3-0.2j
 0,         0.1-0.5j,  0.5-0.7j

 0.5+0.5j, -0.1-0.2j,  0.3-0.6j
 0.5+0.5j,  0.1+0.2j, -0.3+0.6j
 0,         0.3-0.9j,  0.1+0.3j

 0.5+0.5j, -0.1-0.3j,  0.2-0.6j
 0.5+0.5j,  0.1+0.3j, -0.2+0.6j
 0,         0.4-0.8j,  0.2+0.4j

 0.5+0.5j, -0.3-0.4j,  0.3-0.4j
 0.5+0.5j,  0.3+0.4j, -0.3+0.4j
 0,         0.1-0.7j,  0.1+0.7j

 0.5+0.5j, -0.3-0.4j,  0.3-0.4j
 0.5+0.5j,  0.3+0.4j, -0.3+0.4j
 0,         0.5-0.5j,  0.5+0.5j

Unitary 4 × 4 non-real complex matrices

There is no easy formula for $4 \times 4$ non-real unitary matrices as there is for real matrices of the same dimension, but we give some examples here:

  0.1+0.1j, -0.1+0.2j, -0.2-0.2j,  0.2+0.9j
  0.1+0.2j,  0.1-0.1j, -0.2+0.9j, -0.2+0.2j
  0.2+0.2j, -0.2+0.9j,  0.1+0.1j, -0.1-0.2j
  0.2+0.9j,  0.2-0.2j,  0.1-0.2j,  0.1-0.1j

  0.1+0.1j,  0.2+0.2j, -0.1+0.6j, -0.2-0.7j
  0.1+0.2j, -0.2+0.9j,  0.1-0.1j, -0.2+0.2j
  0.2+0.2j, -0.1-0.1j, -0.2+0.7j,  0.1+0.6j
  0.2+0.9j,  0.1-0.2j,  0.2-0.2j,  0.1-0.1j

  0.1+0.1j, -0.2+0.2j, -0.1-0.8j, -0.3+0.4j
  0.1+0.2j, -0.2+0.9j,  0.2+0.2j,  0.1-0.1j
  0.2+0.2j,  0.1-0.1j,  0.3+0.4j, -0.1+0.8j
  0.2+0.9j,  0.1-0.2j, -0.1-0.1j,  0.2-0.2j

  0.1+0.1j, -0.2-0.2j, -0.3+0.7j, -0.4-0.4j
  0.1+0.2j, -0.2+0.9j, -0.2-0.1j, -0.1-0.2j
  0.2+0.2j,  0.1+0.1j,  0.4+0.4j,  0.7-0.3j
  0.2+0.9j,  0.1-0.2j,  0.1-0.2j, -0.2+0.1j

  0.1+0.1j,  0.2+0.2j, -0.1+0.6j, -0.2-0.7j
  0.1+0.2j, -0.2+0.9j,  0.1-0.1j, -0.2+0.2j
  0.2+0.2j, -0.1-0.1j, -0.2+0.7j,  0.1+0.6j
  0.2+0.9j,  0.1-0.2j,  0.2-0.2j,  0.1-0.1j

  0.1+0.1j,  0.2+0.2j, -0.3-0.1j, -0.4+0.8j
  0.1+0.2j, -0.9-0.2j, -0.2-0.1j,  0.1+0.2j
  0.2+0.2j, -0.1-0.1j,  0.4+0.8j, -0.3+0.1j
  0.2+0.9j,  0.2+0.1j,  0.1-0.2j,  0.2-0.1j

  0.1+0.1j, -0.2+0.9j,  0.1-0.2j, -0.2-0.2j
  0.1+0.2j, -0.1+0.2j, -0.1-0.2j,  0.6+0.7j
  0.2+0.2j,  0.2+0.1j,  0.7+0.6j,  0.1+0.1j
  0.2+0.9j,  0.1-0.2j, -0.1-0.2j, -0.1-0.2j

  0.1+0.1j, -0.2+0.9j, -0.1-0.2j,  0.2+0.2j
  0.1+0.2j,  0.2-0.2j,  0.5-0.1j,  0.5+0.6j
  0.2+0.2j, -0.1+0.2j,  0.6+0.5j,  0.1-0.5j
  0.2+0.9j,  0.1-0.1j, -0.2-0.2j, -0.2-0.1j

  0.1+0.1j, -0.2+0.2j, -0.1+0.4j,  0.3+0.8j
  0.1+0.2j,  0.6+0.7j, -0.1-0.2j, -0.2+0.1j
  0.2+0.2j,  0.1-0.1j,  0.8+0.3j, -0.4+0.1j
  0.2+0.9j, -0.1-0.2j, -0.1-0.2j,  0.2-0.1j

  0.1+0.1j, -0.2+0.2j, -0.5+0.2j,  0.5+0.6j
  0.1+0.2j, -0.7+0.6j,  0.2-0.1j, -0.1-0.2j
  0.2+0.2j,  0.1-0.1j,  0.5-0.6j,  0.5+0.2j
  0.2+0.9j,  0.2-0.1j, -0.2+0.1j, -0.1-0.2j

  0.1-0.1j,  0.4+0.1j,  0.6+0.4j, -0.5+0.2j
  0.1+0.2j, -0.4-0.2j,  0.1-0.4j, -0.7-0.3j
  0.2+0.2j,  0.5-0.6j, -0.5+0.1j, -0.2+0.1j
  0.2+0.9j, -0.1+0.1j,  0.2+0.1j,  0.2+0.2j

  0.6+0.7j,  0.2-0.2j, -0.1-0.1j, 0.1-0.2j
  0.2+0.2j, -0.6+0.7j, 0.1+0.2j, 0.1-0.1j
  0.1+0.2j,  0.1-0.1j, 0.4+0.6j, -0.4+0.5j
  0.1+0.1j, -0.1+0.2j, -0.4-0.5j, -0.4+0.6j

  0.6+0.7j,  0.2-0.2j, -0.1-0.2j,  0.1-0.1j
  0.2+0.2j, -0.6+0.7j, -0.1+0.1j,  0.2+0.1j
  0.1+0.2j, -0.1+0.1j,  0.4+0.3j, -0.8-0.2j
  0.1+0.1j,  0.1-0.2j,  0.2+0.8j,  0.3+0.4j

  0.6+0.7j,  0.2-0.1j, -0.1-0.2j, -0.1-0.2j
  0.2+0.2j, -0.6-0.1j,  0.2-0.1j,  0.5+0.5j
  0.1+0.2j, -0.2+0.7j, -0.1+0.4j, -0.4+0.3j
  0.1+0.1j, -0.1-0.2j,  0.3+0.8j,  0.2-0.4j

  0.6+0.7j, -0.1-0.2j,  0.2-0.1j, -0.1-0.2j
  0.2+0.2j,  0.3-0.2j, -0.5+0.2j,  0.1+0.7j
  0.1+0.2j, -0.1+0.8j,  0.2-0.1j,  0.4+0.3j
  0.1+0.1j, -0.1+0.4j, -0.5+0.6j, -0.2-0.4j

  0.6+0.7j,  0.2-0.1j, -0.1-0.2j,  0.2-0.1j
  0.2+0.2j, -0.6-0.1j,  0.5-0.1j, -0.5+0.2j
  0.1+0.2j, -0.2+0.7j, -0.2+0.5j, -0.2-0.3j
  0.1+0.1j, -0.1-0.2j, -0.2+0.6j,  0.2+0.7j

  0.6+0.7j,  0.2-0.1j,  0.2-0.1j, -0.1-0.2j
  0.2+0.2j, -0.6+0.2j, -0.5-0.1j,  0.5-0.1j
  0.1+0.2j,  0.1-0.2j, -0.5+0.6j, -0.2+0.5j
  0.1+0.1j, -0.1+0.7j,  0.2-0.2j, -0.2+0.6j

  0.6+0.7j, -0.2-0.2j, -0.1-0.1j,  0.1-0.2j
  0.2+0.2j,  0.7+0.6j,  0.1+0.2j,  0.1-0.1j
  0.1+0.2j, -0.1-0.1j,  0.4+0.6j, -0.4+0.5j
  0.1+0.1j,  0.2+0.1j, -0.4-0.5j, -0.4+0.6j