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Discover Linear Algebra

Jeremy Sylvestre

Section 41.2 Terminology and notation

quadratic form
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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
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cross terms (in a quadratic formula)
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mixed-variable terms of the form \(a x_i x_j \text{,}\) with \(i \neq j\)
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diagonal quadratic form
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a quadratic form with no cross terms
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level set (at level \(k\) for a multivariable function \(f\))
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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
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principal axes (for symmetric real matrix \(A\))
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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)
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