Matrix principal minors

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. ... 

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. 

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,... 

It is well known in linear algebra that the necessary and sufficient condition for a matrix to be positive-definite is that all principal minors of the determinant are positive. We then immediately find that Cy and ks must be positive ... 

MALDI matrix spectra are characterized by strong molecular and/or quasi-molecular ion signals accompanied by series of matrix (Ma) cluster ions and some more abundant fragment ions. [32] In positive-ion MALDI, [Man-hH]" cluster ions predominate, while [Ma -H] cluster ions are preferably formed in negative-ion MALDI. The principal ion series may be accompanied by [Ma H-alkali] ions and some fragments of minor intensity, e.g., [Ma H-H-H20]. In particular with aprotic matrices, radical ions may predominate. In addition, a continuous background is... 

The IR spectra of the oil and asphaltene neutrals (35) exhibited no significant absorptions in the region 3200-3600 cm-1 except for H20 bands at 3620-3695 cm-1 in the matrix-isolation spectrum of the oil neutrals. Weak absorptions near 1700 cm-1 are indicative of minor amounts of ketones/ aldehydes in both neutral fractions. The absorptions at 2860, 2950, and 3050 cm-1 are ascribable to aliphatic and aromatic CH stretching. The band at 1600 cm-1 is characteristic of aromatic ring C=C. Thus, the oxygen-containing compounds in both neutral fractions are principally composed of ethers. 

Figure 12 The elements of a bivariate variance-covariance matrix lie on the boundary defined by an ellipse. The major axis of the ellipse represents the first pruwipal component, and its minor axis the second principal component...

Radical formation in the polyimides on y-radiolysis under vacuum has been investigated by ESR spectroscopy. At 77 K the radical spectra of the irradiated polyimides are composed principally of two partially overlapping singlets (8,9), one of which decays when the temperature is raised to 200 K. This component has been assigned to anion radicals formed at 77 K by trapping of thermalized electrons in the polymer matrix. The other singlet has been assigned to neutral radicals with the free electron delocalized over several units of the polymer chain. These radicals are thus similar to the radicals formed on UV photolysis. Other minor radical components are also sometimes evident in the spectra, for example in that of Ultem that has been irradiated at 77 K (9). 

Let us now indicate the principal moments of inertia of a molecule (i.e., the eigenvalues of the matrix I) by X 1 s / < 3. The anisometry of the molecule can be expressed in terms of these moments. We can define a simple descriptor in analogy with the eccentricity of planar ellipses. The ellipse s eccentricity is the ratio (M — rn y lM, where M and m, respectively, are the length of the major and minor elliptical axes." Similarly, we can define a molecular eccentricity e in terms of the principal moments of inertia as ... 

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