The characteristic roots of matrices

Parker, W. V.

doi:10.1215/s0012-7094-45-01247-6

Cited by 34 publications

(9 citation statements)

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“…1 We point out that the Loewner partial order on induces a preorder on the LOS matrices , for ~~and~~ implies =~~, but not~= . It only implies~= for some unitary matrix (see, for instance, [3], [4] …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Monotonicity Results for Coherent MIMO Rician Channels

Hoesli

Kim²,

Lapidoth

2005

IEEE Trans. Inform. Theory

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Abstract-The dependence of the Gaussian input information rate on the line-of-sight (LOS) matrix in multiple-input multiple-output (MIMO) coherent Rician fading channels is explored. It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix , or more precisely, monotonic in in the sense of the Loewner partial order. Conversely, it is also demonstrated that this ordering on the LOS matrices is a necessary condition for the uniform monotonicity over all input covariance matrices. This result is subsequently applied to prove the monotonicity of the isotropic Gaussian input information rate and channel capacity in the singular values of the LOS matrix. Extensions to multiple-access channels (MAC) are also provided.Index Terms-Capacity, line of sight, multiple-access channels (MAC), multiple-input multiple-output (MIMO), outage, partial order, preorder, Rician fading.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Monotonicity Results for Coherent MIMO Rician Channels

Hoesli

Kim²,

Lapidoth

2005

IEEE Trans. Inform. Theory

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“…For two given matrices A, B ∈ M n , let E(A, B) be the set of eigenvalues of matrices of the form U * AU +V * BV for some unitary matrices U and V . Wielandt [19] (see also, [3] and [15]) determined the set E(A, B) for two normal matrices A, B ∈ M n . There is not much information about the set E(A, B) for general matrices A, B ∈ M n .…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of the Sum of Matrices from Unitary Similarity Orbits

Li¹,

Poon²,

Sze³

2008

SIAM J. Matrix Anal. & Appl.

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Let A and B be n × n complex matrices. Characterization is given for the set E(A, B) of eigenvalues of matrices of the form U * AU + V * BV for some unitary matrices U and V . Consequences of the results are discussed and computer algorithms and programs are designed to generate the set E (A, B). The results refine those of Wielandt on normal matrices. Extensions of the results to the sum of matrices from three or more unitary similarity orbits are also considered.2000 Mathematics Subject Classification. 15A18.

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“…For an nXn complex matrix, Parker [1]1 has proven that the maximum value, for fixed A, of I(Ax,y)1 (1) subj ect to the constraint llxll=IIYII=l is the maximum singular value, or Hilbert norm, sl(A) of A. In fact, it is clear that the maximum value in (1) occurrs exactly when x is a unit eigenvector in en corresponding to the maximum eigenvalue of A *A, and y = Ax/IIAxll.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, it is clear that the maximum value in (1) occurrs exactly when x is a unit eigenvector in en corresponding to the maximum eigenvalue of A *A, and y = Ax/IIAxll.…”

Section: Introductionmentioning

confidence: 99%