A price model maps market-input changes to price changes. Its adjoint carries a price-weighted sensitivity back to the inputs. That is the algebra behind least-squares normal equations and many gradient calculations. This note defines linear functionals, constructs the adjoint from an inner product, and explains when its matrix is a conjugate transpose. An adjoint preserves a pairing; it does not generally invert the model.
Learning objectives. By the end of this note you should be able to:
- state the defining property of the adjoint operator;
- compute the adjoint of a linear map given its matrix;
- prove the basic algebraic properties of the adjoint;
- explain why the adjoint always exists and is unique.
In this lesson
- Linear functionals
- The adjoint operator
- Properties of the adjoint
Before you start
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