Question

What are the signs of the determinants of the principal submatrices of a negative definite matrix? See Theorem 8.2.5.

Short answer

the solution is

$\left|{\mathrm{A}}^{\left(1\right)}\right|<0,\left|{\mathrm{A}}^{\left(2\right)}\right|>0,\left|{\mathrm{A}}^{\left(3\right)}\right|<0,\left|{\mathrm{A}}^{\left(4\right)}\right|>0,...,$, and so on are the signs of principal submatrices of a negative definite matrix .

Step-by-step solution

Find submatrices of a negative definite matrix

A is a definite negative matrix.

$$\begin{gathered} \Rightarrow {\mathrm{x}}^{\mathrm{T}}\mathrm{Ax}<0\mathrm{for}\mathrm{all}\mathrm{x}\in \mathrm{R},\mathrm{x}\ne 0 \\ \Rightarrow -\left({\mathrm{x}}^{\mathrm{T}}\mathrm{Ax}\right)>0\mathrm{for}\mathrm{all}\mathrm{x}\in \mathrm{R},\mathrm{x}\ne 0 \\ \Rightarrow {\mathrm{x}}^{\mathrm{T}}\left(-\mathrm{A}\right)\mathrm{x}>0\mathrm{for}\mathrm{all}\mathrm{x}\in \mathrm{R},\mathrm{x}\ne 0 \end{gathered}$$

$\Rightarrow -\mathrm{A}$is positive definite matrix

We know that X is a positive definite matrix if and only if det $\left({\mathrm{X}}^{\left(\mathrm{m}\right)}\right)>0;\mathrm{m}=1,2...,\mathrm{n}$from Theorem 8.2.5. So we have $\left({\mathrm{B}}^{\left(\mathrm{m}\right)}\right)>0\mathrm{for}\mathrm{all}\mathrm{m}=1,2,...,\mathrm{n}\mathrm{for}=-\mathrm{A}$for all for . We now have an odd .

for all odd m ,

$$\begin{gathered} \left|{\mathrm{B}}^{\left(\mathrm{m}\right)}\right|>0\mathrm{for}\mathrm{all}\mathrm{odd}\mathrm{m} \\ \Rightarrow \left|{\left(-\mathrm{A}\right)}^{\left(\mathrm{m}\right)}\right|>0 \\ \Rightarrow \left|{\left(-\mathrm{A}\right)}^{\left(\mathrm{m}\right)}\right|>0(\mathrm{since}\mathrm{m}\mathrm{is}\mathrm{odd}) \\ \Rightarrow {\left(-1\right)}^{\mathrm{m}}\left|{\mathrm{A}}^{\left(\mathrm{m}\right)}\right|>0(\mathrm{since}\left|\mathrm{kX}\right|={\mathrm{k}}^{\mathrm{c}}\left|\mathrm{X}\right|\mathrm{for}\mathrm{X}\mathrm{of}\mathrm{order}\mathrm{c}\mathrm{and}\mathrm{scalar}\mathrm{k}) \\ \Rightarrow -\left|{\mathrm{A}}^{\left(\mathrm{m}\right)}\right|>0(\mathrm{since}\mathrm{m}\mathrm{is}\mathrm{odd}) \\ \Rightarrow \left|{\mathrm{A}}^{\left(\mathrm{m}\right)}\right|<0 \end{gathered}$$

Use even matrix function

Similarly, we have for even m.

$\left|{\mathrm{B}}^{\left(\mathrm{m}\right)}\right|>0$for all even

$$\begin{gathered} \Rightarrow \left|{\left(-\mathrm{A}\right)}^{\left(\mathrm{m}\right)}\right|>0 \\ \Rightarrow \left|{\mathrm{A}}^{\left(\mathrm{m}\right)}\right|>0(\mathrm{since}\mathrm{m}\mathrm{is}\mathrm{even}) \end{gathered}$$

The determinants of the principal submatrices with alternating signs of negative(for odd order) and positive(for even order) for a negative definite matrix A are as follows $\left|{\mathrm{A}}^{\left(1\right)}\right|<0,\left|{\mathrm{A}}^{\left(2\right)}\right|>0,\left|{\mathrm{A}}^{\left(3\right)}\right|<0,\left|{\mathrm{A}}^{\left(4\right)}\right|>0,...,$, and so on are the signs of principal submatrices of a negative definite matrix A .

Conclusion

$\left|{\mathrm{A}}^{\left(1\right)}\right|<0,\left|{\mathrm{A}}^{\left(2\right)}\right|>0,\left|{\mathrm{A}}^{\left(3\right)}\right|<0,\left|{\mathrm{A}}^{\left(4\right)}\right|>0,...,$, and so on are the signs of principal submatrices of a negative definite matrix A.