Column-eigenvector

Once we have computed the matrix of column-eigenvectors V we can derive the corresponding row-eigenvectors U using eq. (31.13) ... 

In the column-problem we compute the matrix of generalized column-eigenvectors B and the diagonal matrix of eigenvalues from ... 

Let us take the column eigenvectors x(/l) and place them side by side to form a square matrix X of eigenvectors ... 

Since F is a Hermitian operator, F is a Hermitian matrix and its eigenvectors can be chosen as orthonormal. Let C be the unitary square matrix of column eigenvectors of F ... 

The SVD is generally accepted to be the most numerically accurate and stable technique for calculating the principal components of a data matrix. MATLAB has an implementation of the SVD that gives the singular values and the row and column eigenvectors sorted in order from largest to smallest. Its use is shown in Example 4.3. We will use the SVD from now on whenever we need to compute a principal component model of a data set. 

The set of roots of a polynomial with real coefficients can include conjugate pairs of complex numbers. Thus eigenvalues can be complex, appearing in conjugate pairs. When this happens the corresponding column eigenvectors also form a conjugate pair, as do the rows. 

Clearly the eigenvalue 1 is dominant. Its column eigenvector in the original matrix is a column of all Is19, which means that all the points in this piece of the control polygon will be at the same place in the limit. 

The next column eigenvector is varying linearly, and so the polygon is converging towards a straight line, with points evenly spaced along it. The eigenvalue is 1/2, which means that the density doubles at each step. The first derivative at the limit point is the limit of the first divided difference, which is (B — Z)/2. 

To explore further, we choose our coordinate system so that this straight line is the. x-axis. The subdominant eigencomponent then makes no contribution to y, which is dominated by the third eigencomponent. The column eigenvector looks complicated, but in fact it is just a quadratic variation, with an offset added... 

Going from the scheme to its difference scheme removes the unit eigencom-ponent, and each of the other components has its eigenvalue multiplied by the arity and its (column) eigenvector converted by simple differencing. We can do this as many times as there are a factors in the generating function of the original scheme. 

The column eigenvector of the 1/2 eigenvalue gives the distribution of control points with parameter. 

The first of these has exactly the same eigenvectors as the binary quadratic scheme of the previous question, but eigenvalues 1,1/3,1/9,1/9. Again, the fourth of the column eigenvectors is not polynomial, and the associated eigenvalue is 1/9 and so the Holder continuity is no better than —log3( 1/9) = 1 + 1. 

The second, which gives the neighbourhood of a limit point corresponding to a control point has eigenvalues 1,1/3,1/9, with column eigenvectors ... 

The column eigenvectors are all polynomial, and so there is no constraint on the Holder continuity here. 

The third column eigenvector is non-polynomial, and so the Holder continuity is no higher than —logz 1/6) ps 1 + 0.65. 

Finally, one can show that decays with certain phase distributions cannot arise from the same /V-level system as decays with certain other phase distributions, even though each of the distributions is one of the 2[Pg.286]

Denote the eigenvalues as A, (i= l,2,...,n) then we can define a corresponding column eigenvector, denoted m, for each eigenvalue (an n x 1 matrix) which is defined as the solution of the equation... 

Find the eigenvalues and eigenvectors of the matrix equation given and show the matrix of the column eigenvectors diagonalizes the original matrix. 

Because i = 1,..., N, Equation A.26 represents the N different equations. However, from the rules for matrix multiplication, it is easy to realize that these equations can be collectively represented by one matrix equation. This is accomplished by collecting the column eigenvectors into one N xN matrix. 

---