Vector space/Finite-dimensional/Inner product/Adjoint endomorphism/Matrix/Fact

Matrix description of the adjoint endomorphism

Let $V$ be a finite-dimensional $𝕂$-vector space, endowed with an inner product $⟨-,-⟩$. Let

$\phi :V⟶V$

be an endomorphism, and suppose that it is described by the matrix $M$ with respect to the orthonormal basis $u_1,\dots ,u_n$.

Then the

adjoint endomorphism

$\hat{\phi }$ is described by the matrix ${\bar{M}}^{\text{tr}}$ with respect to this basis .

Proof, Write another proof