Square matrix trace

It is well known that the trace of a square matrix (i.e., the sum of its diagonal elements) is unchanged by a similarity transfonnation. If we define the traces... 

The trace of a square matrix A of dimension n is equal to the sum of the n elements on the main diagonal ... 

The trace Tr A of a square matrix A is defined as the sum of the diagonal elements... 

Other notation used diagB is the diagonal n x n matrix consisting of the diagonal elements of the square matrix B. The trace of B is denoted trB, and the determinant of B is denoted B. The Kronecker product of two matrices is denoted by symbol (g). Other notation will be introduced as needed. 

Two important complex numbers associated to any particular complex linear operator T (on a finite-dimensional complex vector space) are the trace and the determinant. These have algebraic definitions in terms of the entries of the matrix of T in any basis however, the values calculated will be the same no matter which basis one chooses to calculate them in. We define the trace of a square matrix A to be the sum of its diagonal entries ... 

The matrix (2.13) is a square matrix of order n. The elements axva22,...,ann lie on the principal diagonal of A the sum of these elements is called the trace of the square matrix ... 

If we take a square matrix S and form the product T 1ST, where T is any nonsingular square matrix, we have carried out a similarity transformation on S if R = T-lST, then R and S are similar matrices. Similar matrices have equal traces. The proof uses (2.15) ... 

The real parts of the eigenvalues are negative, and the perturbations will decay in time as Figure 12.3 illustrates. When the value of B is 2.4 then the oscillations are sustainable. Figure 12.3b and 12.3d show the state-space plot of concentrations Cx and CY for the different values of B. Regardless of whether the eigenvalues are real or complex, the steady state is stable to small perturbations if the two conditions tr[J] < 0 and dct. / > 0 are satisfied simultaneously. Here, tr is the trace and det is the determinant of the square matrix J. 

Here, the symbol Tr[X] (read trace of X ) means the sum of the diagonal elements of the square matrix [X] with elements Xjk. 

The character, defined only for a square matrix, is the trace of the matrix, or the sum of the numbers on the diagonal from upper left to lower right. For the C2v point group, the following characters are obtained from the preceding matrices ... 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

One way to derive this is to eliminate the coefficients for the upper set between Eqs. (19-19). This leads to an eigenvalue equation for eigenvalues of the matrix <5o /) -hZj, W y Wyi,. The sum of eigenvalues (sum of E ) over this lower set is exactly equal to the trace of this matrix. Similarly, the sum of E over the set is exactly equal to the trace of the squared matrix. One can also write these sums as sums over , = — ITj H- A,-, noticing that A,- has second-order and fourth-order terms, and solve for the sum of Eq. (19-20). It is also readily confirmed that this is correct to fourth order for the special case of only two coupled states by expanding the exact solution, >/lTf2 + VVj, in Wi2-... 

Equation (10.45) can be differentiated with respect to time and combined with Equation (10.44) to obtain the relationship between the multicomponent copolymer composition and CSTR monomer compositions. It can be seen that trace of a square matrix is a scalar quantity. 

Associated with each square matrix is a determinant (see below). If the determinant of a square matrix vanishes, the matrix is said to be singular. A singular matrix has no inverse. The trace of a square matrix is the sum of the diagonal elements of the matrix ... 

As a consequence, PCs are sort in decreasing variance order and considering the algebraic property of the conservation of the trace, that is, for any nonsingular square matrix, B, given its diagonal D, it holds trace(B) = trace(D), the sum of the eigenvalues equals the total variance of the X matrix ... 

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