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Unitary Similarity, Normal Matrices, and Spectral Theory
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
Normal transformations are those which have an orthogonal basis of eigenvectors and, thus, can be represented by diagonal matrices relative to an orthonormal basis. The class of normal transformations includes Hermitian, skew-Hermitian, and unitary transformations; studying normal matrices leads to a more unified understanding of all of these special types of transformations. Often, results that are discovered first for Hermitian matrices can be generalized to the class of normal matrices. Since normal matrices are unitarily similar to diagonal matrices, things that are obviously true for diagonal matrices often hold for normal matrices as well; for example, the singular values of a normal matrix are the absolute values of the eigenvalues. Normal matrices have two important properties — diagonalizability and an orthonormal basis of eigenvectors — that tend to make life easier in both theoretical and computational situations.
Compact operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
In the previous chapter, we introduced and studied compact operators on Banach spaces. In this chapter, we will restrict attention to compact operators on Hilbert spaces, where, due to the unique geometry of Hilbert spaces, we will be able to say much more. The culmination of this chapter is the so-called spectral theorem for compact normal operators on a Hilbert space. The spectral theorem is the infinite dimensional version of the familiar fact that every normal matrix is unitarily diagonalizable.
Exact Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
There are many ways to understand when a matrix A is normal or non-normal. The matrix A is normal if and only if it is diagonalizable and its different eigenspaces are orthogonal to each other.Every matrix is unitarily similar to an upper-triangular matrix. Hence, up to unitary similarity, every normal matrix is diagonal, and every non-normal matrix is upper-triangular, but not diagonal. For example, matrices of the form λ110λ2 are never normal.Normal matrices (i.e., those satisfying A*A=AA*, where A* is the conjugate transpose of A) include those that are unitary, Hermitian, orthogonal, or symmetric.Suppose for some vector x we have A2x=0 but Ax≠0. Then A is not normal. In particular, any nonzero nilpotent operator is non-normal.
Properties of semi-conjugate gradient methods for solving unsymmetric positive definite linear systems
Published in Optimization Methods and Software, 2023
Na Huang, Yu-Hong Dai, Dominique Orban, Michael A. Saunders
If A is a normal matrix, we have , where X is a unitary matrix and with is the diagonal matrix containing the eigenvalues of A. Then If for all , we have Then from Theorem 3.1 we know that SWI is convergent provided