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The Hilbert matrix is also totally positive (meaning that the determinant of every submatrix is positive). The Hilbert matrix is an example of a Hankel matrix. It is also a specific example of a Cauchy matrix. The determinant can be expressed in closed form, as a special case of the Cauchy determinant. The determinant of the n × n Hilbert ...
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1] It is a result of studies of linear algebra and the solutions of systems of linear equations and their ...
The class of normal operators is well understood. Examples of normal operators are unitary operators: N* = N −1; Hermitian operators (i.e., self-adjoint operators): N* = N; skew-Hermitian operators: N* = −N; positive operators: N = MM* for some M (so N is self-adjoint). A normal matrix is the matrix expression of a normal operator on the ...
The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. [89] In a Hilbert space H, an orthonormal basis is a family {e k} k ∈ B of elements of H satisfying the conditions: Orthogonality: Every two different elements of B are orthogonal: e k, e j = 0 for all k, j ∈ B with k ≠ j.
This example can be expanded to R 3. In even higher dimensions, this can be extended to the Givens rotation. Reflections, like the Householder transformation. times a Hadamard matrix. In general, any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary.
The last property given above shows that if one views as a linear transformation from Hilbert space to , then the matrix corresponds to the adjoint operator of . The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.