Classify a quadratic form depending on a parameter

Question

Let $Q:\mathbb{R^3}\to\mathbb{R}$ be a quadratic form, $Q(x,y,z)=x^2+2axy+2xz+z^2$ with $a \in \mathbb{R}$.

Classify the form depending on $a$.

The first thing I did was to find the matrix of the quadratic form:

Let $A$ be that matrix:

$A=\begin{pmatrix} 1 & a & 1 \\ a & 0 & 0 \\ 1 & 0 & 1 \\ \end{pmatrix}$

And then I tried to find the characteristic polynomial, and that's where I got stuck:

$|A-\lambda I|=\begin{vmatrix} 1-\lambda & a & 1 \\ a & \lambda & 0 \\ 1 & 0 & 1-\lambda \\ \end{vmatrix}$

Using Sarrus' rule I got that $\chi _A=-\lambda^3+2\lambda^2+a^2\lambda-a^2$.

I only know that when $a=0$ then $\chi _A=-\lambda^3+2\lambda^2=\lambda^2(-\lambda+2)$, so the quadratic form would be positive semidefinite, but I don't know how to manipulate the polynomial enough to classify it when $a>0$ or $a<0$.

Is there an easier way to do this?

Answer 1

Maybe you can use Principal Minors diagonal form: $$q_J(x,y,z)=D_1x^2+(D_2/D_1)y^2+(D_3/D_2)z^2.$$

In your case, $D_1=1$, $D_2=\begin{vmatrix}1&a\\a&0\end{vmatrix}=-a^2$ and $D_3={\rm det}(A)=-a^2$.

If $a\ne0$, then $q_J(x,y,z)=x^2-a^2y^2+z^2$ and, thus, the quadratic form is indefinite.

If $a=0$, $q(x,y,z)=(x+z)^2$ which is clearly positive semidefinite.

Answer 2

Unnecessary.

Just observe:

$Q(x,y,z)=(x+z)^2+\dfrac{a}{2}\big( (x+y)^2-(x-y)^2 \big)$

and conclude that the for is positive semidefinite when $a = 0$ and non-definite otherwise.

Answer 3

Illustrating a cleverness-free method, using $a = 8, a = -6, a = 0$ When $a \neq 0,$ the eigenvalue signs are $++-$

The matrix identities below, called $Q^T DQ = H,$ work out (when $a \neq 0$) to $$ (x+ay+z)^2 - (ay+z)^2 + z^2 \; . \; $$ Notice that, when $a=0,$ the one-line expression is misleading, two terms cancel and there is just $(x+z)^2.$ Meanwhile, when $a=0,$ the matrix product below has undefined terms. The algorithm, you see, works to make $\det Q = \pm 1,$ although Sylvester's Inertia allows for any nonsingular $Q$

$$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ a & 1 & 0 \\ 1 & \frac{ 1 }{ a } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - a^2 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & a & 1 \\ 0 & 1 & \frac{ 1 }{ a } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & a & 1 \\ a & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$

Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
$$ H = \left( \begin{array}{rrr} 1 & 8 & 1 \\ 8 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$ $$ D_0 = H $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$

$$ H = \left( \begin{array}{rrr} 1 & 8 & 1 \\ 8 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$

==============================================

$$ E_{1} = \left( \begin{array}{rrr} 1 & - 8 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrr} 1 & - 8 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrr} 1 & 8 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & - 64 & - 8 \\ 1 & - 8 & 1 \\ \end{array} \right) $$

==============================================

$$ E_{2} = \left( \begin{array}{rrr} 1 & 0 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{2} = \left( \begin{array}{rrr} 1 & - 8 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{2} = \left( \begin{array}{rrr} 1 & 8 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{2} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 64 & - 8 \\ 0 & - 8 & 0 \\ \end{array} \right) $$

==============================================

$$ E_{3} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & - \frac{ 1 }{ 8 } \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{3} = \left( \begin{array}{rrr} 1 & - 8 & 0 \\ 0 & 1 & - \frac{ 1 }{ 8 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{3} = \left( \begin{array}{rrr} 1 & 8 & 1 \\ 0 & 1 & \frac{ 1 }{ 8 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{3} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 64 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$

==============================================

$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ - 8 & 1 & 0 \\ 0 & - \frac{ 1 }{ 8 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 8 & 1 \\ 8 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - 8 & 0 \\ 0 & 1 & - \frac{ 1 }{ 8 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 64 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 8 & 1 & 0 \\ 1 & \frac{ 1 }{ 8 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 64 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 8 & 1 \\ 0 & 1 & \frac{ 1 }{ 8 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 8 & 1 \\ 8 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

$$ H = \left( \begin{array}{rrr} 1 & - 6 & 1 \\ - 6 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$ $$ D_0 = H $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$

$$ H = \left( \begin{array}{rrr} 1 & - 6 & 1 \\ - 6 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$

==============================================

$$ E_{1} = \left( \begin{array}{rrr} 1 & 6 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrr} 1 & 6 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrr} 1 & - 6 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & - 36 & 6 \\ 1 & 6 & 1 \\ \end{array} \right) $$

==============================================

$$ E_{2} = \left( \begin{array}{rrr} 1 & 0 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{2} = \left( \begin{array}{rrr} 1 & 6 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{2} = \left( \begin{array}{rrr} 1 & - 6 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{2} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 36 & 6 \\ 0 & 6 & 0 \\ \end{array} \right) $$

==============================================

$$ E_{3} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & \frac{ 1 }{ 6 } \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{3} = \left( \begin{array}{rrr} 1 & 6 & 0 \\ 0 & 1 & \frac{ 1 }{ 6 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{3} = \left( \begin{array}{rrr} 1 & - 6 & 1 \\ 0 & 1 & - \frac{ 1 }{ 6 } \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{3} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 36 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$

==============================================

$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 6 & 1 & 0 \\ 0 & \frac{ 1 }{ 6 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - 6 & 1 \\ - 6 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 6 & 0 \\ 0 & 1 & \frac{ 1 }{ 6 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 36 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ - 6 & 1 & 0 \\ 1 & - \frac{ 1 }{ 6 } & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & - 36 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & - 6 & 1 \\ 0 & 1 & - \frac{ 1 }{ 6 } \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & - 6 & 1 \\ - 6 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

$$ H = \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$ $$ D_0 = H $$ $$ E_j^T D_{j-1} E_j = D_j $$ $$ P_{j-1} E_j = P_j $$ $$ E_j^{-1} Q_{j-1} = Q_j $$ $$ P_j Q_j = Q_j P_j = I $$ $$ P_j^T H P_j = D_j $$ $$ Q_j^T D_j Q_j = H $$

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

==============================================

$$ E_{1} = \left( \begin{array}{rrr} 1 & 0 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) $$ $$ P_{1} = \left( \begin{array}{rrr} 1 & 0 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; Q_{1} = \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) , \; \; \; D_{1} = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) $$

==============================================

$$ P^T H P = D $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ - 1 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & - 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) $$ $$ Q^T D Q = H $$ $$\left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) = \left( \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array} \right) $$

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=