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Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the <i>D</i>-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a <i>D</i>-invariant polynomial subspace <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula> in terms of Cartesian tensors, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula> is a subspace with a maximal total degree equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. For an arbitrary homogeneous polynomial <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mstyle scriptlevel="1" displaystyle="false"><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mstyle></msup></semantics></math></inline-formula> of total degree <i>k</i> in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mstyle scriptlevel="1" displaystyle="false"><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mstyle></msup></semantics></math></inline-formula> can be rewritten as the inner products of a <i>k</i>th order symmetric Cartesian tensor and <i>k</i> column vectors of indeterminates. We show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mstyle scriptlevel="1" displaystyle="false"><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mstyle></msup></semantics></math></inline-formula> can be determined by all polynomials of a total degree one in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula>. Namely, if we treat all linear polynomials on the basis of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula> as a column vector, then this vector can be written as a product of a coefficient matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>A</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula> and a column vector of indeterminates; our main result shows that the <i>k</i>th order symmetric Cartesian tensor corresponds to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mstyle scriptlevel="1" displaystyle="false"><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mstyle></msup></semantics></math></inline-formula> is a product of some so-called relational matrices and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>A</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></semantics></math></inline-formula>.