Thesis (Ph. D.)--Columbia University, 1931. / Vita.

Thesis (Ph. D.)--University of Virginia, 1907.

Thèse--Faculté des sciences de Paris, 1915.

Thesis (Ph. D.)--Johns Hopkins University, 1915. / Vita. From American journal of mathematics, vol. 39, no. 2 (April, 1917)

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 116).

Thesis--Wisconsin. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, 1922. / Biography. "A Dissertation, submitted to the Faculty of the Ogden Graduate School of Science in candidacy for the Degree of Doctor of Philosophy, Department of Mathematics. "Private edition, distributed by the University of Chicago Libraries, Chicago, Illinois." "Reprinted from Transactions American mathematical society, vol. XXIV, no. 2, September, 1922." Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, 1920. / Vita. "Private edition, distributed by the University of Chicago libraries." Also available on the Internet. Also issued online.

In this thesis two problems concerning linear transformations on Mn, the algebra of n-square matrices over the complex numbers, are considered. The first is the determination of the structure of those transformations which map non-singular matrices to non-singular matrices; the second is the determination of the structure of those transformations which, for some positive integer r, preserve the sum of the r x r principal subdeterminants of each matrix. In what follows, we use E to denote this sum, and the phrase "direct product" to refer to transformations of the form T(A) = cUAV for all A in Mn or T(A) = cUA'V for all A in Mn where U, V are fixed members of Mn and c is a complex number. The main result of the thesis is that both non-singularity preservers and Er-preservers, if r ≥ 4, are direct products. The cases r=1,2,3 are discussed separately. If r=1, it is shown that E₁ preservers have no significant structure. If r=2, it is shown that there are two types of linear transformations which preserve E₂, and which are not direct products. Finally, it is shown that these counter examples do not generalize to the case r=3. These results and their proofs will also be found in a forthcoming paper by M. Marcus and JR. Purves in the Canadian Journal of Mathematics, entitled Linear Transformations of Algebras of Matrices: Invariance of the Elementary Symmetric Functions. / Science, Faculty of / Mathematics, Department of / Graduate