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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\$.