Characteristic roots and field of values of a matrix

Parker, W. V.

doi:10.1090/s0002-9904-1951-09477-x

Cited by 15 publications

(3 citation statements)

References 25 publications

(3 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is clear that this analysis is applicable to any system obeying equation [2] and such that agj/lx1 = dj < 0. What is more, arguments such as those leading from equation [5] to equation [9] can still be used 8 Personal communication.…”

Section: Ia-xii = Id-xi'i[a-x-r(d-xit)-1c] = Id-xi'i H(a-xi) = O [15]mentioning

confidence: 99%

“…If P and C are the number of positive and complex roots, respectively, then n =N+ P+ C [8] and from equations [7,8] n-N= P+C= p+1 [9] where p + 1 is an odd integer not less than 3. Since complex roots, if any, must occur in conjugate pairs, C is either zero or an even integer not exceeding p, and equation [9] shows that P cannot be zero but must be an odd integer. Thus we see that H(A) > 0 implies the existence of at least one positive root and hence instability.…”

mentioning

confidence: 99%

“…Let A1 = aj + i,31, where i = v-1, and aj and I, are real, be any root of A. It is a known theorem (8,9) that if M and m are the maximum and minimum roots, respectively, of B, then m < a. < M, j = 1, 2,*,n [11] In fact this theorem is sufficiently simple to prove here (see Note II).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Applications of Results from Linear Kinetics to the Hodgkin-Huxley Equations

Hearon¹

1964

Biophysical Journal

View full text Add to dashboard Cite

On the basis of its role in the analysis of mammillary compartmental systems, a matrix with non-zero elements in the first row, first column, and along the main diagonal and with zero elements elsewhere is called a mammillary matrix. It is pointed out that such matrices occur in a variety of biological problems including the linearized Hodgkin-Huxley equations (H-H). In considering whether such a linear system exhibits stability (all roots of the matrix with negative real parts) it is of interest to seek conditions, expressible in a simple manner in terms of the matrix elements, which lead to stability or instability. For the case when the diagonal elements, with the possible exception of the first, are negative (a condition physically guaranteed for the space-clamped axon) simple criteria for instability and stability are formulated in terms of the matrix elements. These criteria are derived by extending previous results from linear kinetics through appeal to a classical matrix theorem without recourse to the characteristic polynomial. The relation of these mathematical results to the work of Chandler, FitzHugh, and Cole on the space-clamped axon is discussed. The results are in no way restricted by the order of the matrix (which is four for the H-H equations) and other possible applications are noted.

show abstract