TRD Decomposition of A Locus Ellipsoid - Yang, Wei-Chi; Morante, Antonio (The electronic journal of mathematics & technology, 2024)
We extend the ideas of finding sheared maps discussed in [10], and continue a matrix decomposition called TRD decomposition which has an interesting geometric interpretation. Let M be a three times three invertible matrix with real entries. The matrix M can be written as product of three matrices T, R and D, M = TRD, where D is a diagonalizable matrix with two equal eigenvalues, R is an orthogonal matrix and finally T is a shear matrix. The product TRD is corresponding to a series of linear transformations that send the unit sphere to the same ellipsoid that M does. The decomposition for a general ellipsoid has been discussed in [6]. In this paper, the decomposition is applied on a locus ellipsoid [L.sub.E] ([summation]), resulted from a linear transformation [L.sub.E] that is applied on an ellipsoid [summation], which is discussed in ([9]) and ([8]). Moreover, [L.sub.E] ([summation]) can be represented by a positive definite M. we adopt a different approach when decompose M into TRD. We relate the given ellipsoid to an ellipsoid that is in its standard form through a transition matrix. Next, we apply the SVD decomposition on a sheared ellipsoid to obtain the final decomposition for the given locus ellipsoid [L.sub.E] ([summation]).