What is the name of the operation $v^T H v$?

Question

To find out how something is called is the one thing I don't know how to use Google for.

If I have a vector $v$ and a symmetric matrix $H$, then the operation $v^T H v$ gives me some sort of projection of the matrix $H$ on the vector $v$. For example if $H$ is the Hessian, the operation gives me the second derivative in the direction of $v$.

What is the name of this operation?

Answer 1

• $(u,v) \mapsto u^T H v$ is called a bilinear form. It is called a symmetric bilinear form if $H$ is symmetric. It is called an inner product if $H$ is symmetric positive definite.
• $v \mapsto v^T H v$ is called a quadratic form.
• You might also be interested in $v \mapsto \frac{v^T H v}{v^T v}$ which is called a Rayleigh quotient.