Sets of complex numbers associated with a matrix

Parker, W. V.

doi:10.1215/s0012-7094-48-01560-9

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A Note On Matrix Solutions To a = Xy-yx Charles R Johnson11

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1955

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Cited by 27 publications

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“…Neither Lemma 1 nor Lemma 2 holds over arbitrary fields. Lemma 1 does hold over the real field [4], but it is easy to construct examples to show that an algebraically closed field is not a superfluous assumption in Lemma 2. In particular Lemma 2 does not hold over the…”

Section: A Note On Matrix Solutions To a = Xy-yx Charles R Johnson1mentioning

confidence: 99%

A note on matrix solutions to $A=XY-YX$

Johnson¹

1974

Proc. Amer. Math. Soc.

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Abstract.It is known that a square matrix A can be written as a commutator XY-YX'xi and only if Tr(/1)=0. In this note it is shown further that for a fixed A the spectrum of one of the factors may be taken to be arbitrary while the spectrum of the other factor is arbitrary as long as the characteristic roots are distinct. The distinctness restriction on one of the factors may not in general be relaxed.

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Section: A Note On Matrix Solutions To a = Xy-yx Charles R Johnson1mentioning

confidence: 99%

A note on matrix solutions to $A=XY-YX$

Johnson¹

1974

Proc. Amer. Math. Soc.

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“…,[4]. If Tr(A)=0, then A is unitarily equiralent to a matrix B=(bij) with bij=O, i=1, -*, n.C. R. JOHNSON[February LEMMA 2(FRIEDLAND [2]).…”

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A Note on Matrix Solutions to A = XY - YX

Johnson¹

1974

Proceedings of the American Mathematical Society

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“…Let (u, v) be the unitary inner product and set f(z) = (Cz, z). The range of values of / for ||z|| =(z, z)1/2 = l is the field of values [1] of C and we denote it by F(C). Set A = (C+C*)/2, B = (C-C*)/2i, <b(z) = (Az, z), and f(z) = (Bz, z).…”

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confidence: 99%