Latent root

J. C. Gower. Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53 325, 1966. 

E. Vigneau, D. Bertrand and E.M. Qannari, Application of latent root regression for calibration in near-infrared spectroscopy. Comparison with principal component regression and partial least squares. Chemometr. Intell. Lab. Syst., 35 (1996) 231-238. 

The solution x = 0 is excluded. There may be several different column vectors x, each with a different value of A, and each satisfying the equation. The numbers A are called the latent roots or eigenvalues of A. The vectors x are the latent solutions or eigenfunctions of A. The equation, written out in full, is a homogeneous system of linear equations, and will have a solution, other than x = 0, if and only if... 

It is commonly asserted that the latent roots of this Jacobian matrix are all real and negative as a consequence of the principle of microscopic reversibility. However, a discussion of this would take us outside the bounds of our subject. 

McClelland, B.J. (1971). Properties of the Latent Roots of a Matrix The Estimation of n-Elec-tron Energies. J.Chem.Phys., 54,640-643. 

Eigenvalues, which are also sometimes called latent roots or characteristic roots, are important in determining the stability of a matrix to inversion and eigenvalues/ eigenvectors play an important role in many aspects of multivariate statistical analysis like principal component analysis. If X is a square symmetrical matrix then X can be decomposed into... 

The method defines a new set of orthogonal axes called eigenvectors or latent vectors. For instance, the first eigenvector is the direction of maximum spread of the data in terms of n-dimensional space. It is a best fit line in n-dimensional space and the original data can be projected onto this vector using the first set of principal component coordinates. The variance of these coordinates is the first eigenvalue or latent root, and is a measure of the spread in the direction of the first eigenvector. 

H1] S. J. Hammarling. Latent Roots and Latent Vectors. University of Toronto Press, Toronto. 1970, pp. 30-2. 

A matrix product of the form A" HA is called a similarity transformation on H. If A is orthogonal, then AHA is a special kind of similarity transformation, called an orthogonal transformation. If A is unitary, then A HA is a unitary transformation on H. There is a physical interpretation for a similarity transformation, which will be discussed in a later chapter. For the present, we are concerned only with the mathematical definition of such a transformation. The important feature is that the eigenvalues, or latent roots, of H are preserved in such a transformation (see Problem 9-5). 

Show that, if a matrix has any latent roots equal to zero, it has no inverse. 

If a latent root is zero, then the product of latent roots is zero. But this product is the value of the determinant of the matrix. If the determinant of the matrix is zero, there is no inverse. 

Latent roots The deficiencies in the management systems or the management approaches that allow human errors to continue unchecked... 

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