I will define a nice singular-value decomposition as one where the orthogonal matrices $U$ and $V$ have finite decimal representations with at most a few digits to the right of the decimal point. This makes it much easier to demonstrate this in class, but such examples are very rare for smaller matrices, and it is not necessarily obvious how to find such examples. In some cases, this author believes the descriptions below give are exhaustive.
There are only two $2 \times 2$ orthogonal matrices that we are likely to make use of:
$U_1 = \begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix}$
U1 = [0.6, 0.8
0.8, -0.6]
$U_2 = \begin{pmatrix} 0.28 & 0.96 \\ 0.96 & -0.28 \end{pmatrix}$
U2 = [0.28, 0.96
0.96, -0.28]
Similarly, there is only one useful $3 \times 3$ matrix that is orthogonal:
$U_3 = \begin{pmatrix} 0.6 & 0.48 & 0.64 \\ 0.8 & -0.36 & -0.48 \\ 0 & -0.8& 0.6 \end{pmatrix}$
U3 = [0.6, 0.48, 0.64
0.8, -0.36, -0.48
0, -0.8, 0.6]
Fortunately, there are many $4 \times 4$ orthogonal matrices:
$U_4 = \begin{pmatrix} 0.5 & 0.5 & 0.5 & 0.5 \\ 0.5 & -0.5 & -0.5 & 0.5 \\ 0.5 & 0.5 & -0.5 & -0.5 \\ 0.5 & -0.5 & 0.5 & -0.5 \end{pmatrix}$
U4 = [0.5, 0.5, 0.5, 0.5
0.5, -0.5, -0.5, 0.5
0.5, 0.5, -0.5, -0.5
0.5, -0.5, 0.5, -0.5]
$U_5 = \begin{pmatrix} 0.2 & 0.4 & 0.4 & 0.8 \\ 0.4 & -0.2 & -0.8 & 0.4 \\ 0.4 & 0.8 & -0.2 & -0.4 \\ 0.8 & -0.4 & 0.4 & -0.2 \end{pmatrix}$
U5 = [0.2, 0.4, 0.4, 0.8
0.4, -0.2, -0.8, 0.4
0.4, 0.8, -0.2, -0.4
0.8, -0.4, 0.4, -0.2]
$U_6 = \begin{pmatrix} 0.1 & 0.1 & 0.7 & 0.7 \\ 0.1 & -0.1 & -0.7 & 0.7 \\ 0.7 & 0.7 & -0.1 & -0.1 \\ 0.7 & -0.7 & 0.1 & -0.1 \end{pmatrix}$
U6 = [0.1, 0.1, 0.7, 0.7
0.1, -0.1, -0.7, 0.7
0.7, 0.7, -0.1, -0.1
0.7, -0.7, 0.1, -0.1]
$U_7 = \begin{pmatrix} 0.1 & 0.5 & 0.5 & 0.7 \\ 0.5 & -0.1 & -0.7 & 0.5 \\ 0.5 & 0.7 & -0.1 & -0.5 \\ 0.7 & -0.5 & 0.5 & -0.1 \end{pmatrix}$
U7 = [0.1, 0.5, 0.5, 0.7
0.5, -0.1, -0.7, 0.5
0.5, 0.7, -0.1, -0.5
0.7, -0.5, 0.5, -0.1]
$U_8 = \begin{pmatrix} 0.1 & 0.3 & 0.3 & 0.9 \\ 0.3 & -0.1 & -0.9 & 0.3 \\ 0.3 & 0.9 & -0.1 & -0.3 \\ 0.9 & -0.3 & 0.3 & -0.1 \end{pmatrix}$
U8 = [0.1, 0.3, 0.3, 0.9
0.3, -0.1, -0.9, 0.3
0.3, 0.9, -0.1, -0.3
0.9, -0.3, 0.3, -0.1]
$U_9 = \begin{pmatrix} 0.5 & 0.5 & 0.1 & 0.7 \\ 0.5 & 0.5 & -0.1 & -0.7 \\ 0.5 & -0.5 & -0.7 & 0.1 \\ 0.5 & -0.5 & 0.7 & -0.1 \end{pmatrix}$
U9 = [0.5, 0.7, 0.1, 0.5
0.5, -0.7, -0.1, 0.5
0.5, 0.1, -0.7, -0.5
0.5, -0.1, 0.7, -0.5]
We will now look at examples of these, but you will note that it is necessary to ensure the singular values are distinct, and in some cases, non-zero, depending on the matrices. The distinct singular values ensure distinct eigenvalues, and therefore solving for a basis for the null space ensures that the student finds at least a scalar multiple of the vector in question.
It is always possible to create a block-diagonal matrix using smaller orthogonal matrices. For example, you could have a $3 \times 3$ matrix with either $U_1$ or $U_2$ together with the diagonal entry $1$; or you could create a $4 \times 4$ matrix that has any combination, including repetitions, of $U_1$ or $U_2$ on the diagonal, or one of those two matrices with the other two diagonal entries being $1$.
To give one example,
$U_{10} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0.6 & 0.8 \\ 0 & 0.8 & -0.6 \end{pmatrix}$
U10 = [1, 0, 0
0, 0.6, 0.8
0, 0.8, -0.6]
These have a minimum number of digits beyond the decimal point in the matrix $A = UDV^\textrm{T}$:
These have a minimum number of digits beyond the decimal point in the matrix $A = UDV^\textrm{T}$:
nice eigendecompositions integer matrices with integer eigenvalues integer matrices with integer singular values integer-normed vectors integer-normed complex vectors geometric sequences with nice norms