Self-adjoint linear transformation

A linear transformation of a Euclidean or unitary space that coincides with its adjoint linear transformation. A self-adjoint linear transformation in a Euclidean space is also called symmetric, and in a unitary space, Hermitian. A necessary and sufficient condition for the self-adjointness of a linear transformation of a finite-dimensional space is that its matrix $A$ in an arbitrary orthonormal basis coincides with the adjoint matrix $A^*$, that is, it is a symmetric matrix (in the Euclidean case), or a Hermitian matrix (in the unitary case). The eigenvalues of a self-adjoint linear transformation are real (even in the unitary case), and the eigenvectors corresponding to different eigenvalues are orthogonal. A linear transformation of a finite-dimensional space $L$ is self-adjoint if and only if $L$ has an orthonormal basis consisting of eigenvectors; in this basis the transformation can be described by a real diagonal matrix.

A self-adjoint linear transformation $A$ is non-negative (or positive semi-definite) if $(Ax,x)\geq0$ for any vector $x$, and positive definite if $(Ax,x)>0$ for any $x\neq0$. For a self-adjoint linear transformation in a finite-dimensional space to be non-negative (respectively, positive-definite) it is necessary and sufficient that all its eigenvalues are non-negative (respectively, positive), or that the corresponding matrix is positive semi-definite (respectively, positive-definite). In this case there is a unique non-negative self-adjoint linear transformation $B$ satisfying the condition $B^2=A$, that is, $B$ is the square root of the self-adjoint linear transformation $A$.

References

[a1]	P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958)