<p>The sub-defect of a doubly substochastic matrix <i>A</i>, denoted by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(sd(A) = \lceil n-\mathrm sum(A)\rceil \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>⌈</mo> <mi>n</mi> <mo>-</mo> <mi mathvariant="normal">s</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⌉</mo> </mrow> </math></EquationSource> </InlineEquation>, is defined as the minimum number of rows and columns required to be added to transform the doubly substochastic matrix into a doubly stochastic matrix. Here, <i>n</i> signifies the matrix size, and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathrm sum(A)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">s</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> represents the sum of all entries of <i>A</i>. In this paper, we provide the full characterization of all linear operators under which the sub-defect is preserved.</p>