The trace of a matrix

The trace of a square matrix AnXn, denoted tr 4, is the sum of its diagonal elements 

The trace of a scalar is the scalar itself. Since the inner (dot) product of two vectors xn and y is a scalar, we can write 

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It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

This result is often stated in words that, the trace of a matrix is independent of the representation of the matrix, i. e. 

Invariance of the trace of a matrix under unitary transformation corresponds to the invariance of phase density under canonical transformation in classical theory. 

Now comes the very principle of the principal component analysis. A total variance is now defined as the trace of the matrix Sx or, using a property of the trace of a matrix product given in Section 2.2... 

Since the trace of a matrix is invariant upon rotation, we get the sum of the n eigenvalues 8j as the trace of the correlation matrix. The dimensionless total variance is therefore... 

Because the trace of a matrix is independent of the coordinate system, matrices representing operations that have the same effect in different coordinate systems must have the same trace. It is possible to use this fact to abbreviate the character tables. For example, consider the long and short versions of the character table of ... 

Based on the above similarity transform, we can now show that the trace of a matrix (i.e., the sum of its diagonal elements) is independent of the representation in which the matrix is formed, and, in particular, the trace is equal to the sum of the eigenvalues of the matrix. The proof of this theorem proceeds as follows ... 

The trace of a matrix which represents an element of a group (or an operation of a point group) is called a character and is usually given the symbol X- X(R) is thus the character of the operation R in the representation which has matrices D(R)> i.e. 

As a preliminary for deriving a useful expression, we note that the trace of a matrix is invariant under unitary transformation and that Tr(AB) = Tr (BA). For any unitary matrix 11, it follows that 0 = (LflP) = UlP + UUP and that Tr(U W + UPU) = 0, the prime indicating the energy derivative. Therefore, we have... 

The trace of a matrix is defined as the sum of the diagonal elements, and q is given as the rank of Rj. Please note in the above derivations, as in RAFA, that R, must be a rank 1 matrix representing the pure standard. 

The trace of a matrix is the sum of the terms along the principal diagonal, transcendental number... 

Now we to concentrate on the properties of the two-dimensional lattice, the space-time lattice. The partition function, Eq. (133), shows that there is coupling only in the time direction and only between nearest-neighbor time slices. This allows us to use the statistical mechanics technique of writing the partition function Z of the finite system as the trace of a matrix T to the power Nt. 

The sum of the eigenvalues of a matrix equals the trace of the matrix the trace of a matrix being the sum of the elements on the principal, or main, diagonal. 

This is true regardless of the perturbation mechanism (J-dependent or J-independent matrix elements) because the trace of a matrix is representation invariant the sum of the basis function (i.e., deperturbed) energies is equal to the sum of the eigenvalues. Since the approximately unperturbed Bmain and Emain(O) values are usually known from a relatively perturbation-free portion of the band, the constants for the perturbing state can be inferred from Eqs. (4.3.10) and (4.3.11). If the average energy plot shows any deviation from linearity, this implies either an incorrect line identification or an additional perturber. 

In any representation of a symmetry group, the trace of a matrix is called the character of the corresponding operator for that representation. 

The advantage of this high-temperature expansion is that we need not determine the individual energy levels but can use the theorem of the invariance of the trace of a matrix. For example, to find the value of e2 we square the Hamiltonian and pick out only the diagonal terms. On summing these terms over all possible values, many of the sums cancel. For example, the zero-field splitting Hamiltonian of the form... 

Since the trace of a matrix (C = AB) is invariant under a unitary transformation we can write... 

The parameters range in the interval 0 and 1 tr characterizes the trace of a matrix and I is the identity matrix. 

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