Discover Linear Algebra

Section 41.2 Terminology and notation

quadratic form

a multivariable polynomial formula so that each term has degree two; that is, if the variables are $x_1,x_2,\dotsc,x_n\text{,}$ each term in the formula is of the form

$$\begin{equation*} a x_i x_j \text{,} \end{equation*}$$

where each $a$ is an arbitrary constant, and where $i = j$ is allowed to produce a true quadratic term

cross terms (in a quadratic formula)

mixed-variable terms of the form $a x_i x_j \text{,}$ with $i \neq j$

diagonal quadratic form

a quadratic form with no cross terms

level set (at level $k$ for a multivariable function $f$)

the collection of points in $\R^n$ that satisfy $f(x_1,x_2,\dotsc,x_n) = k\text{;}$ also called a level curve for a piecewise-continuous function in two variables or a level surface for a piecewise-continuous function in three variables

principal axes (for symmetric real matrix $A$)

an orthonormal basis $\basisfont{B}$ for $\R^n$ so that the transition matrix $\ucobmtrx{B}{S}$ orthogonally diagonalizes $A$ (where $\basisfont{S}$ is the standard basis)