Geodesic
On matrices that admit unitary reduction to band form
Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 871-880

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It is proved in the paper that any low rank perturbation of a Hermitian matrix is unitarily reducible to band form. Moreover, if a normal matrix is unitarily reducible to band form, then any of its rank one perturbations is unitarily reducible as well. 
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     author = {Kh. D. Ikramov and L. \'El'zner},
     title = {On matrices that admit unitary reduction to band form},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
     volume = {64},
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     year = {1998},
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     url = {http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a7/}
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Kh. D. Ikramov; L. Él'zner. On matrices that admit unitary reduction to band form. Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 871-880. http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a7/