Diagonalizable Matrices

A matrix is diagonalizable if it is similar to a diagonal matrix. Diagonalizable matrices are of particular interest since their matrix functions can be easily computed. Let us see how a... 

It is easy to see a matrix has a closed orbit if and only if it is diagonalizable. Hence the set of closed orbits can be identified with the set of eigenvalues. On the other hand, a matrix B with [B, 5I] = 0 (i.e. a normal matrix) can be diagonalizable by a unitary matrix. Hence the quotient space is also identified with the set of eigenvalues. The identification can be seen directly in this example. 

If the result of a similarity transformation is to produce a diagonal matrix (see Appendix A.4-l(b)), then the process is called diagonaliz-ation. If the matrices A and B can be diagonalized by the same matrix, then A and B commute. 

Remark. The detailed balance relation (4.2) or (6.1) asserts that the matrix W is virtually symmetric and will be seen in the next section to guarantee that W can be diagonalized. The relation (6.14) is also a property of W but does not by itself guarantee diagonalizability, and wil be referred to as extended detailed balance . The relations (6.12) and (6.13) are not properties of W but relate the transition probabilities in one system to those in another system. They will therefore not be honored with the name detailed balance. The extended detailed balance property will be important in XI.4. 

As mentioned before, the properties (V.2.5), which characterize a W-matrix, are not sufficient to guarantee that there exists a matrix S such that S 1WS is diagonal. The additional detailed balance property (4.2) or (6.1), however, makes W symmetric in a certain sense and thereby diagonalizable, see (6.15). In this section we develop the consequences. 

If W is a finite matrix, linear algebra tells us that this can in fact be asserted when W is symmetric. If W is an operator in an infinite-dimensional space the mathematical conditions are considerably more complicated, but as a rule of thumb one may also regard any symmetrical operator as diagonaliz-able - as is customary in quantum mechanics. It will now be shown that the detailed balance property guarantees that the operator W is symmetrical. We adopt the notation of a continuous range. 

Some matrices are diagonalizable over C in the sense that they have a basis of - possibly complex - eigenvectors and these matrices can therefore be diagonalized by matrix similarity. For example, this is so for all n by n matrices with n distinct eigenvalues since eigenvectors for different eigenvalues are linearly independent. Other matrices such... 

Specifically for normal matrices, defined by the matrix equation A A = AA, this implies orthogonal diagonalizability for all normal matrices, such as symmetric (with AT = A e R" "), hermitian A = A Cn,n), orthogonal (ATA = /), unitary (A A = /), and skew-symmetric (AT = —A) matrices. 

A matrix is diagonalizable if it is equivalent to a diagonal matrix D. The characteristic equation of A is invariant under a similarity transformation, for... 

In the general case, if A is not a normal matrix, then it is not necessarily diagonalizable. However, it is diagonalizable if the characteristic equation has n distinct roots. 

The presence of the continuum, coupled with the SVCA, results in a complex-I mmetiTC g matrix. Such matrices are diagonalizable using complex-orthogonal... 

Normality is a convenient test for diagonalizability every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is also normal (but finding the desired transform requires much more work than simply testing to see whether the matrix is normal). d Changing the signs simultaneously of certain trios (e.g. g, g )) of elements of g will leave G untouched. Our... 

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n x n matrix g separable if the subalgebra k[p] of End(/c") is separable. We have of course k[g] k[X]/p(X) where p(X) is the minimal polynomial of g. Separability then holds iff k[g] k = /qg] a fc(Y]/p(.Y) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k, and g is diagonalizable over k,. 

Ixt k be algebraically closed, G an algebraic matrix group. Show G is unipotent iff all elements of finite order have order divisible by char (fc). [Use Kolchin s theorem to reduce to the abelian case, and look at diagonalizable matrix groups.]... 

Provided the matrix fV is diagonalizable (which we shall assume throughout this Chapter cf. also Appendix 2 and Section IV), the system of differential equations can be decoupled by means of the transformation... 

Functions of a diagonalizable matrix can be evaluated by an alternative procedure. If [A] represents a diagonal matrix... 

In general, we can evaluate any function of a diagonalizable matrix [yl] by the above procedure. Thus,... 

An alternative to the methods described above can be used if the coefficient matrix is diagonalizable. Consider, once again, the matrix differential equation and its associated initial condition... 

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