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 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.
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References
[a1] | P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958) |
Self-adjoint linear transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-adjoint_linear_transformation&oldid=33469