Non positive-definiteness of some binomial matrices

Question

Experience shows that the following matrix $$ A_4 = \begin{pmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 3 & 4 & 0 \\ 1 & 3 & 6 & 0 & 0 \\ 1 & 4 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{pmatrix}, $$ which could be called a binomial matrix, or a Pascal matrix, is not positive definite, along the like matrices $A_1$, $A_2$, $A_3$. Is this a theorem valid for all $n$?

Same question for $$ B_4 = \begin{pmatrix} 1 & c_1 & c_2 & c_3 & c_4\\ c_1 & 2c_2 & 3c_3 & 4c_4 & 0 \\ c_2 & 3c_3 & 6c_4 & 0 & 0 \\ c_3 & 4c_4 & 0 & 0 & 0 \\ c_4 & 0 & 0 & 0 & 0 \\ \end{pmatrix}, $$ and like matrices, where the $c_i$'s are arbitrary reals.

Answer 1

the original question, used in Sylvester's Law of Inertia. The displayed matrix products fit with the usual method of repeated "completing the square". In the $n$ by $n$ case, about half the coefficients may be taken to be $1,$ the rest negative.

$$ Q^T D Q = H $$

$$\left( \begin{array}{rr} 1 & 0 \\ 1 & 1 \\ \end{array} \right) \left( \begin{array}{rr} 1 & 0 \\ 0 & - 1 \\ \end{array} \right) \left( \begin{array}{rr} 1 & 1 \\ 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rr} 1 & 1 \\ 1 & 0 \\ \end{array} \right) $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

$$ Q^T D Q = H $$

$$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & - 1 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 2 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 1 & 1 \\ 0 & 1 & - 1 \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 2 & 0 \\ 1 & 0 & 0 \\ \end{array} \right) $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

$$ Q^T D Q = H $$

$$\left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & - 1 & - \frac{ 1 }{ 5 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 5 & 0 \\ 0 & 0 & 0 & - \frac{ 9 }{ 5 } \\ \end{array} \right) \left( \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & - 1 \\ 0 & 0 & 1 & - \frac{ 1 }{ 5 } \\ 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 0 \\ 1 & 3 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

$$ Q^T D Q = H $$

$$\left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 \\ 1 & 3 & - 7 & 1 & 0 \\ 1 & - 1 & 1 & - \frac{ 9 }{ 59 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & - 59 & 0 \\ 0 & 0 & 0 & 0 & - \frac{ 96 }{ 59 } \\ \end{array} \right) \left( \begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & - 1 \\ 0 & 0 & 1 & - 7 & 1 \\ 0 & 0 & 0 & 1 & - \frac{ 9 }{ 59 } \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrr} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 0 \\ 1 & 3 & 6 & 0 & 0 \\ 1 & 4 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ \end{array} \right) $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$

$$ Q^T D Q = H $$

$$\left( \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 2 & 1 & 0 & 0 & 0 \\ 1 & 3 & 3 & 1 & 0 & 0 \\ 1 & 4 & - 9 & - \frac{ 14 }{ 19 } & 1 & 0 \\ 1 & - 1 & 1 & \frac{ 1 }{ 19 } & - \frac{ 107 }{ 833 } & 1 \\ \end{array} \right) \left( \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 19 & 0 & 0 \\ 0 & 0 & 0 & 0 & - \frac{ 1666 }{ 19 } & 0 \\ 0 & 0 & 0 & 0 & 0 & - \frac{ 1250 }{ 833 } \\ \end{array} \right) \left( \begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 & 4 & - 1 \\ 0 & 0 & 1 & 3 & - 9 & 1 \\ 0 & 0 & 0 & 1 & - \frac{ 14 }{ 19 } & \frac{ 1 }{ 19 } \\ 0 & 0 & 0 & 0 & 1 & - \frac{ 107 }{ 833 } \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 & 5 & 0 \\ 1 & 3 & 6 & 10 & 0 & 0 \\ 1 & 4 & 10 & 0 & 0 & 0 \\ 1 & 5 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) $$

$$ \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc $$