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The kernel trick evaluates inner products of feature map outputs through $k(x,z)=⟨\varphi (x),\varphi (z)⟩$ without explicitly constructing those vectors. An algorithm expressed entirely through these inner products can therefore operate on nonlinear derived coordinates using only evaluations of a positive-semidefinite kernel on the original inputs. A classifier linear in the derived coordinates can have a nonlinear boundary in the original coordinates.

The polynomial kernel gives a finite, fully visible example: $k(x,z)=(x^{T}z{)}^2$ corresponds to $\varphi (x)=(x_1^2,\sqrt{2}{x}_1{x}_2,x_2^2{)}^T$. The XOR separation by a product feature below makes the interaction coordinate $x_1{x}_2$ useful, while dual prediction with a kernel shows how to predict without calculating it explicitly.

An arbitrary score called a similarity need not be a positive-semidefinite kernel. Symmetry alone, even with nonnegative entries, is insufficient: the symmetric matrixhas $a^{T}Sa=-2$ for $a=(1,-1{)}^T$, so it cannot be a Gram matrix of real inner products. A positive-semidefinite kernel requires a positive semidefinite matrix for every finite choice of inputs, not just the observed training data.^[ref] The existing Moore-Aronszajn theorem connects this requirement to a Reproducing kernel Hilbert space; this example needs only finite linear algebra.

For $x,z\in {\mathbb{R}}^d$, a polynomial kernel is the positive-semidefinite kernel $k(x,z)=(x^{T}z+c{)}^r$, where $r$ is a positive integer and $c\ge 0$. The case $c=0$ uses monomials of total degree $r$; $c>0$ also includes lower degrees. The feature map weights matter: listing each distinct monomial once with coefficient one generally gives a different inner product.^[ref]

For the homogeneous quadratic case in two dimensions, take the scaled quadratic feature mapThe inner product identity is explicit:The factor $\sqrt{2}$ accounts for the two equal cross terms in the square. Thus an inner product of three derived coordinates can be obtained from an inner product of the two input coordinates followed by squaring. The kernel trick uses this equality inside an algorithm; the feature map explains which nonlinear functions the algorithm can represent.