Matrix latent roots

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... 

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. 

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