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A quadratic form is a homogeneous polynomial of degree two in a number of variables. Example: $Q(x, y) = ax^2 + bxy + cy^2$, where $a, b,$ and $c$ are constants.

Quadratic forms can be written in matrix form as $Q(x) = x^T A x$, where $x$ is a column vector of variables and $A$ is a symmetric matrix. Example: For $Q(x, y) = 3x^2 + 2xy + 3y^2$, the matrix $A$ would be $\begin{bmatrix}3 & 1 \\ 1 & 3\end{bmatrix}$.

The Principal Axes Theorem states that any real symmetric matrix representing a quadratic form can be diagonalized by an orthogonal change of coordinates. Example: $Q(x, y) = x^2 - xy + y^2$ can be diagonalized to $Q(u, v) = u^2 + v^2$ by an appropriate rotation.

A quadratic form is positive definite if it takes only positive values (except at the origin), and negative definite if it takes only negative values (except at the origin). Example: $x^2 + y^2$ is positive definite, $-x^2 - y^2$ is negative definite.

A quadratic form is positive semi-definite if it takes only non-negative values and negative semi-definite if it takes non-positive values. Example: $x^2$ is positive semi-definite.

The discriminant of a quadratic form in two variables $ax^2 + bxy + cy^2$ is given by $D = b^2 - 4ac$. It indicates the nature of the conic section represented by the quadratic form. Example: If the discriminant is negative, the conic is an ellipse.

The eigenvalues of the symmetric matrix associated with a quadratic form give information about the form's definiteness. A quadratic form is positive definite if all the eigenvalues are positive. Example: The eigenvalues of $\begin{bmatrix}3 & 1 \\ 1 & 3\end{bmatrix}$ are positive.

Quadratic forms can have three or more variables, typically written as $ax^2 + by^2 + cz^2 + dxy + exz + fyz$. Example: $2x^2 + 2y^2 + 3z^2 - xy - yz$.

A quadratic form can sometimes be reduced to a sum of squares by an orthogonal transformation. Example: $Q(x, y) = x^2 - xy + y^2$ becomes $Q(u, v) = u^2 + v^2$ after diagonalization.

The rank of a quadratic form is the rank of its corresponding symmetric matrix. Example: The quadratic form represented by the matrix $\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}$ has rank 1.

The signature of a quadratic form is the number of positive eigenvalues minus the number of negative eigenvalues of the corresponding matrix. Example: For the matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$, the signature is 0.

Real quadratic forms have coefficients that are real numbers. Example: $Q(x, y) = 3x^2 + 4y^2$ has real coefficients 3 and 4.

A necessary and sufficient condition for a quadratic form to be positive definite is that all its leading principal minors of the corresponding matrix are positive. Example: For $\begin{bmatrix}1 & 2 \\ 2 & 8\end{bmatrix}$, the leading principal minors are 1 and 6, both positive.

A quadratic form can be seen as a dot product of a vector with itself pre-multiplied and post-multiplied by a symmetric matrix. Example: $\begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = x^2 + y^2$.

The canonical form of a quadratic form is achieved by a linear change of variable that simplifies the equation to a sum of squares with coefficients +1 or -1. Example: $2x^2 - 3y^2$ is already in canonical form.

Cross product terms in quadratic forms are the terms involving the product of different variables. Example: In $Q(x, y, z) = x^2 + y^2 + z^2 + 2xy - 2xz$, the terms $2xy$ and $-2xz$ are cross product terms.

Transforming a quadratic form involves changing variables to simplify the form or to make its properties more apparent. Example: Transforming $x^2 - xy + y^2$ by rotating axes to remove the $xy$ term.

Quadratic forms often appear in optimization problems, where finding the maximum or minimum involves completing the square or using matrix calculus. Example: Minimizing $Q(x, y) = 4x^2 + 4xy + y^2$.

Quadratic forms can be graphically represented as conic sections such as ellipses, hyperbolas, or parabolas in the xy-plane. Example: $Q(x, y) = x^2 + 4xy + 4y^2$ represents an ellipse.