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Definition. Given a square matrix ${\bf{A}}=[a_{ij}] \in {\mathbb{C}^{n \times n}}$, the submatrix ${{\bf{A}}_{{i_1},{i_2},...{i_k}}}$ is formed by retaining the $({i_1},{i_2},...{i_k})$-th rows and columns of $\bf{A}$.

For example, ${{\bf{A}}_1} = {a_{11}}$, ${{\bf{A}}_{2,3}} = \left[ {\begin{array}{*{20}{c}}{{a_{22}}}&{{a_{23}}}\\{{a_{32}}}&{{a_{33}}}\end{array}} \right]$, and ${{\bf{A}}_{1,2,...n}} = {\bf{A}}$.

Question. What is the sufficient or necessary condition for the sum of submatrix determinant of the same size $\left| {\sum\limits_{k{\rm{ }}\;{\rm{fixed}}} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right|$ to monotonically decreases (non-increasing) as size $k$ rises?

Example 1: rank 1 matrix. Any rank 1 matrix has the above property, since $\left| {{{\bf{A}}_{{i_1},{i_2},...{i_k}}}} \right|=0$ when $k \ge 1$. Obviously, only $\left|\det ( {{{\bf{A}}_{{i_1}}}}) \right|,{i_1} = 1...n$ can have nonzero value.

However, tests show that other matrices may also enjoy such property, as long as $a_{ij}\ll 1$?

Example 2: matrix with small elements. $${\bf{A}} = \left[ {\begin{array}{*{20}{r}}{0.144916}&{0.21851}&{ - 0.0008}&{0.024152}\\{ - 0.2079}&{ - 0.20095}&{0.001077}&{ - 0.02566}\\{ - 0.03753}&{ - 0.07695}&{0.023376}&{0.008451}\\{ - 0.12973}&{ - 0.05353}&{0.001961}&{0.005781}\end{array}} \right]$$

$$ \begin{array}{l}\left| {\sum\limits_{k = 1} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {0.144916 - 0.20095 + 0.023376 + 0.005781} \right| = 0.0269\\\left| {\sum\limits_{k = 2} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {\det {{\bf{A}}_{1,2}} + \det {{\bf{A}}_{1,3}} + \det {{\bf{A}}_{1,4}} + \det {{\bf{A}}_{2,3}} + \det {{\bf{A}}_{2,4}} + \det {{\bf{A}}_{3,4}}} \right| = 0.0166\\\left| {\sum\limits_{k = 3} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {\det {{\bf{A}}_{1,2,3}} + \det {{\bf{A}}_{1,2,4}} + \det {{\bf{A}}_{1,3,4}} + \det {{\bf{A}}_{2,3,4}}} \right| = {\rm{0}}{\rm{.0007}}\\\left| {\sum\limits_{k = 4} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {\det {\bf{A}}} \right| = 6 \times {10^{ - 6}}\end{array} $$ Tests show that for matrices with small elements (<<1), the above property is usually satisfied.

However, when the elements are large, the above property may still holds (see next example).

Example 3: matrix with large elements.

$${\bf{A}} = \left[ {\begin{array}{*{20}{r}}{ - 18.5678}&{ - 24.1590}&{0.1273}&{ - 2.4468}\\{ - 0.4104}&{ - 0.4921}&{0.0027}&{ - 0.0497}\\{ - 26.9197}&{ - 35.1067}&{0.1863}&{ - 3.5524}\\{10.0499}&{13.1214}&{ - 0.0683}&{1.3283}\end{array}} \right]$$

$$\begin{array}{l}\left| {\sum\limits_{k = 1} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = 17.5\\\left| {\sum\limits_{k = 2} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {\det {{\bf{A}}_{1,2}} + \det {{\bf{A}}_{1,3}} + \det {{\bf{A}}_{1,4}} + \det {{\bf{A}}_{2,3}} + \det {{\bf{A}}_{2,4}} + \det {{\bf{A}}_{3,4}}} \right| = 0.878\\\left| {\sum\limits_{k = 3} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {\det {{\bf{A}}_{1,2,3}} + \det {{\bf{A}}_{1,2,4}} + \det {{\bf{A}}_{1,3,4}} + \det {{\bf{A}}_{2,3,4}}} \right| = {\rm{0}}{\rm{.0006}}\\\left| {\sum\limits_{k = 4} {\det {{\bf{A}}_{{i_1},{i_2},...{i_k}}}} } \right| = \left| {\det {\bf{A}}} \right| = 3 \times {10^{ - 6}}\end{array}$$

In this example, the matrix ${\bf{A}}$ can be viewed as the sum of a large rank-1 matrix and a small-valued matrix, which I suppose contributes to the property.

But I don't know if there is any effective method to check whether a random matrix satisfies the above property? Does it has something to do with the singular values? or any sorts of matrix decomposition? Can anyone give any clues? Thanks!

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