scholarly journals Latent Roots of Tri-Diagonal Matrices

1961 ◽  
Vol 44 ◽  
pp. 5-7
Author(s):  
F. M. Arscott
Keyword(s):  
Latent Roots

The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models

1975 ◽  
Vol 7 (02) ◽  
pp. 264-282 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Reproductive Structure ◽  
Variable Size ◽  
New Approach ◽  
Age Structured ◽  
Drift Models ◽  
Latent Roots

The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.

scholarly journals On the Latent Roots of Certain Matrices

1928 ◽  
Vol 1 (2) ◽  
pp. 135-138 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Fundamental Theorem ◽  
Present Note ◽  
Principal Diagonal ◽  
Characteristic Determinant ◽  
Elementary Divisors ◽  
The Right ◽  
Latent Roots

In the present note certain known theorems on the latent roots of matrices are deduced from the fundamental theorem that a matrix A can be expressed in the form PQP-1, where P is non-singular and Q has zero elements everywhere to the left of the principal diagonal, and the latent roots of A in the diagonal. [The presence or absence of non-zero elements to the right of the diagonal is known to depend on the nature of the “elementary divisors” of the “characteristic determinant” of A, but in what follows these will not concern us.]

XII.—Clebsch-Aronhold Symbols and the Theory of Symmetric Functions

1951 ◽  
Vol 63 (2) ◽  
pp. 155-173
Author(s):  
H. W. Turnbull ◽  
A. H. Wallace
Keyword(s):  
Bilinear Form ◽  
Fundamental Theorem ◽  
Symmetric Functions ◽  
Projective Invariants ◽  
Invariant Matrices ◽  
Latent Roots ◽  
Square Matrix

SynopsisA square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.

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