$Q(x, y) = ax^2 + 2bxy + cy^2$ (the most general quadratic form on $\mathbb{R}^2$).

Associated bilinear form: $B((x_1, y_1), (x_2, y_2)) = ax_1 x_2 + b(x_1 y_2 + x_2 y_1) + cy_1 y_2$.

Gram matrix: $M = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$.

• $Q = x^2 + y^2$: $M = I$ (positive definite).
• $Q = x^2 - y^2$: $M = \operatorname{diag}(1, -1)$ (indefinite).
• $Q = 2xy$: $M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ (indefinite, $\det = -1$).
• $Q = x^2$: $M = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ (degenerate, rank $1$).

$Q(x, y, z) = x^2 + 2y^2 + 3z^2 + 2xy - 4xz$. The matrix is:

$M = \begin{pmatrix} 1 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & 3 \end{pmatrix}$.

The coefficient of $x_i x_j$ (with $i \neq j$) is $2m_{ij}$, so $m_{12} = 1$ (half of the coefficient $2$ of $xy$) and $m_{13} = -2$ (half of $-4$).

$Q = ax^2 + 2bxy + cy^2$ with $M = \begin{pmatrix} a & b \\ b & c \end{pmatrix}$.

• Positive definite iff $a > 0$ and $\det M = ac - b^2 > 0$.
• Negative definite iff $a < 0$ and $ac - b^2 > 0$.
• Indefinite iff $ac - b^2 < 0$.
• Semi-definite (degenerate) iff $ac - b^2 = 0$.

Examples:

• $Q = x^2 + y^2$: $a = 1, b = 0, c = 1$. $\det = 1 > 0$, $a > 0$. Positive definite.
• $Q = x^2 - y^2$: $\det = -1 < 0$. Indefinite.
• $Q = x^2 + 2xy + y^2 = (x + y)^2$: $\det = 0$. Positive semi-definite (but not positive definite).

$M = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 4 \end{pmatrix}$.

Leading principal minors: $\Delta_1 = 2 > 0$, $\Delta_2 = 6 - 1 = 5 > 0$, $\Delta_3 = \det M = 2(12 - 1) - 1(4) = 18 > 0$.

All positive, so $M$ is positive definite.

$Q(x, y) = 2x^2 + 4xy + 5y^2$.

$= 2(x^2 + 2xy) + 5y^2 = 2(x + y)^2 - 2y^2 + 5y^2 = 2(x + y)^2 + 3y^2$.

Set $u = x + y$, $v = y$: $Q = 2u^2 + 3v^2$ (sum of two positive squares, so positive definite).

Matrix check: $M = \begin{pmatrix} 2 & 2 \\ 2 & 5 \end{pmatrix}$, $\det = 6 > 0$, $a = 2 > 0$. Positive definite ✓.

$Q(x, y, z) = x^2 + 2xy + 2y^2 + 2yz + z^2$.

$= (x + y)^2 + y^2 + 2yz + z^2 = (x + y)^2 + (y + z)^2$.

Set $u = x + y$, $v = y + z$, $w = z$: $Q = u^2 + v^2$.

The form has rank $2$ (only $2$ squares). Signature $(2, 0)$ with rank $2$ out of $3$ variables, so it is positive semi-definite but degenerate.

$Q(x, y) = x^2 - 4xy + y^2$.

$= (x - 2y)^2 - 4y^2 + y^2 = (x - 2y)^2 - 3y^2$.

Set $u = x - 2y$, $v = y$: $Q = u^2 - 3v^2$ (indefinite, signature $(1, 1)$).

• $Q = x^2 + y^2 + z^2$: signature $(3, 0)$ (positive definite).
• $Q = x^2 + y^2 - z^2$: signature $(2, 1)$ (indefinite, the "light cone" form).
• $Q = -x^2 - y^2 - z^2$: signature $(0, 3)$ (negative definite).
• $Q = x^2 - y^2$: signature $(1, 1)$ (hyperbolic form).
• $Q = x^2 + y^2$: on $\mathbb{R}^3$, this has signature $(2, 0)$ and rank $2$ (degenerate).

The second-order Taylor expansion of $f(x)$ at a critical point $x_0$ is:

$f(x_0 + h) \approx f(x_0) + \frac{1}{2} h^T H h,$

where $H$ is the Hessian matrix. The nature of the critical point depends on the quadratic form $Q(h) = h^T H h$:

• $Q$ positive definite: local minimum.
• $Q$ negative definite: local maximum.
• $Q$ indefinite: saddle point.

For $f(x, y) = x^2 + xy + y^2$ at $(0, 0)$: $H = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$, eigenvalues $3, 1 > 0$. Local minimum ✓.

The conic $ax^2 + 2bxy + cy^2 + dx + ey + f = 0$ is classified by the quadratic part $Q = ax^2 + 2bxy + cy^2$:

• $ac - b^2 > 0$: ellipse (or circle if $a = c$ and $b = 0$).
• $ac - b^2 < 0$: hyperbola.
• $ac - b^2 = 0$: parabola.

$x^2 + y^2 = 1$ (circle): $a = c = 1, b = 0$, $ac - b^2 = 1 > 0$. $x^2 - y^2 = 1$ (hyperbola): $ac - b^2 = -1 < 0$. $x^2 = y$ (parabola): $a = 1, b = c = 0$, $ac - b^2 = 0$.

The question "which integers are sums of two squares?" is equivalent to asking for which $n$ the equation $Q(x, y) = x^2 + y^2 = n$ has integer solutions.

Fermat's theorem: a prime $p$ is a sum of two squares iff $p = 2$ or $p \equiv 1 \pmod{4}$.

The form $Q(x, y) = x^2 + y^2$ has discriminant $-4$. Forms of different discriminants represent different sets of integers. This is the beginning of the theory of binary quadratic forms (Gauss).

The moment of inertia of a rigid body about an axis $\hat{n}$ is $I = \hat{n}^T \mathcal{I} \hat{n}$, a quadratic form in the direction $\hat{n}$. The inertia tensor $\mathcal{I}$ is a $3 \times 3$ positive semi-definite symmetric matrix.

The principal axes are the eigenvectors of $\mathcal{I}$, and the principal moments of inertia are the eigenvalues.

For a uniform cube of side $a$ and mass $m$: $\mathcal{I} = \frac{ma^2}{6}I_3$ (isotropic), so $I = \frac{ma^2}{6}$ regardless of the axis direction.

$Q_1 = x^2 + y^2$ and $Q_2 = 2x^2 + 3y^2$: both have signature $(2, 0)$ and rank $2$. They are equivalent over $\mathbb{R}$ (set $u = \sqrt{2}x$, $v = \sqrt{3}y$: $Q_2 = u^2 + v^2 = Q_1$).

$Q_1 = x^2 - y^2$ and $Q_2 = 2xy$: both have signature $(1, 1)$. They are equivalent (set $u = x + y$, $v = x - y$: $2xy = u^2/2 - v^2/2$... after scaling, same signature).

$Q_1 = x^2 + y^2$ and $Q_2 = x^2 - y^2$: signatures $(2, 0)$ and $(1, 1)$. Not equivalent.