Further Applications of Jordan Form

This chapter starts by introducing the notion of root subspaces for quaternion matrices. These are basic invariant subspaces, and the chapter proves in particular that they enjoy the Lipschitz property with respect to perturbations of the matrix. Another important class of invariant subspaces are the one-dimensional ones, i.e., generated by eigenvectors. Existence of quaternion eigenvalues and eigenvectors is proved, which leads to the Schur triangularization theorem (in the context of quaternion matrices) and its many consequences familiar for real and complex matrices. Jordan canonical form for quaternion matrices is stated and proved (both the existence and uniqueness parts) in full detail. The chapter also discusses various concepts of determinants for square-size quaternion matrices. Several applications of the Jordan form are given, including functions of matrices and boundedness properties of systems of differential and difference equations with constant quaternion coefficients.

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WITHDRAWN: THE JORDAN FORM

Linear Algebra and its Applications ◽

10.1016/b978-0-12-673660-1.50013-7 ◽

1980 ◽

pp. 359-366

Author(s):

GILBERT STRANG

Keyword(s):

Jordan Form

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Matrix exponential—Jordan form

Power Electronic System Design ◽

10.1016/b978-0-323-88542-3.00079-0 ◽

2021 ◽

pp. 367

Keyword(s):

Matrix Exponential ◽

Jordan Form

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Unitary Transformations

Canadian Journal of Mathematics ◽

10.4153/cjm-1954-008-3 ◽

1954 ◽

Vol 6 ◽

pp. 69-72 ◽

Cited By ~ 5

Author(s):

B. E. Mitchell

Keyword(s):

Canonical Form ◽

Unitary Transformation ◽

General Pattern ◽

Jordan Form ◽

Unitary Matrix ◽

Complex Matrices ◽

Triangular Form ◽

Unitary Transformations

We consider the problem of finding a unique canonical form for complex matrices under unitary transformation, the analogue of the Jordan form (1, p. 305, §3), and of determining the transforming unitary matrix (1, p. 298, 1. 2). The term “canonical form” appears in the literature with different meanings. It might mean merely a general pattern as a triangular form (the Jacobi canonical form (8, p. 64)). Again it might mean a certain matrix which can be obtained from a given matrix only by following a specific set of instructions (1). More generally, and this is the sense in which we take it, it might mean a form that can actually be described, which is independent of the method used to obtain it, and with the property that any two matrices in this form which are unitarily equivalent are identical.

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