Principal minors

Since the coefficients a can be interpieted as sums of principal minors of A(G), it is easy to check that (i) 0i = 0, (ii) 02 — the number of edges in G, and (hi) 03 = twice the number of triangles in G. A more powerful version of this result will be presented and used in chapter 5. 

We arrive at equation (5.106) by applying this result to all principal minors (each of which is the determinant of a subgraph of of size i ).B... 

If B— [bij] is an N xN matrix in which bu equals the degree of vertex i, bij = —1 if vertices i and j are adjacent and bij = 0 otherwise, then the number of spanning tree of G is equal to the determinant of any principal minor of B [hararybO]. The extremes occur for totally disconnected graphs that have no spanning trees and thus a complexity of zero, and for complete graphs of order N that contain the maximum possible number of distinct trees on N vertices. ... 

The two principal minors (the two diagonal elements] must be positive because p and p are both positive, and the determinant of V2W... 

A necessary and sufficient condition for > 0 is that all its principal minors be nonnegative... 

Proof. A sufficient condition for J, evaluated at to have eigenvalues with negative real parts is that if the oflF-diagonal elements are replaced by their absolute values, then the determinants of the principal minors alternate in sign (Theorem A.11). 

The principal minors are those determinants whose upper left element is flu and have contiguous rows and columns. For example, the first principal minor, d, is just flu- The second is given by... 

The loading of the structure is mechanically very simple since it consists in an initial isotropic stress field related to the dead weight. Concerning the mechanical boundary conditions, a zero normal stress is prescribed on the free boundaries of the pattern (wall of the drifts and well, and ground surface), and the symmetry planes are characterized by a zero normal displacement. Before excavation, the rock mass is supposed to be in a compressive stress state, and the principal minor stress 03, indicator of maximum compression, is equal to CTh (and 0i=O2=o =ayy). This stress increases (in absolute value) with depth, from -1.1 MPa at the top of the wells to -1.6 MPa at its base, and to -3.1 MPa in lower limit of the model. The excavation of drifts and wells causes a disturbance of this initial stress field (see fig.3). It is noticed that, apart... 

Near the well, the principal minor stress Oj is maximum in absolute value (-38 MPa) and its negative value corresponds to compression (see fig.5). Its distribution is relatively comparable with that of the field of temperature. The principal major stress Oi shows traction in the zones where its values are positive (0 to 3 MPa). They are found close to surfaces like the walls of the two galleries, in other words free and not heated surfaces. Lastly, with regard to the intermediate stress 02, it is a compressive one but of a rather average intensity if one does not take account of the localized stresses in the comer of the higher gallery (it is pointed out that the stress is theoretically infinite at this location). 

The matrices K and K are similar matrices they have the same eigenvalues and principal minors [18]. 

Positive and negative definiteness can also be checked using the leading principal minor test, although semidefiniteness cannot be verified in this manner. Using standard matrix notation, let... 

Show that for a general 3x3 matrix, A, the trace, sum of the principal minors i.e., the three minors formed by crossing out the rows and columns of the diagonal elements), and the determinant are invariant under similarity transformations. 

Then, according to the known theorem valid for such positive definite matrices [134, Sects.13.5-13.6], [148, Sect.1.29], its principal minors must be equivalently positive. Writing these determinants as jacobians and using (as follows from (4.203), (4.348))... 

If S S is to be negative for all possible variations 8li and 5V, then the matrix S must be negative definite or equivalently, (-S) must be positive definite. The conditions under which S is negative definite are given by a theorem from linear algebra it is necessary and sufficient that the principal minors of S satisfy the following inequalities [5] ... 

The above procedure can be repeated to obtain the stability criteria for multicomponent mixtures. For a mixture of C components, the criterion is still (8.3.4) in which S is the (C + 2)2 matrix of second derivatives analogous to (8.3.5). The fluid is stable to small disturbances when S is negative definite that is, when odd-order principal minors of S are negative and simultaneously those of even order are positive. The reduction of those minors to economical forms is a tedious exercise that can often be alleviated by posing the criteria in terms of G or A rather than S. 

One way to determine the definiteness of a quadratic form is to determine the signs of its principal minors. In any square matrix A, the principal minors are the determinants I M,-1 formed from the first i rows and i columns of A. For example, for the 2x2 matrix in (B.7.2),... 

A square matrix of order n has n prindpal minors. Then, a quadratic form is positive definite if all its principal minors are positive,... 

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