20.2 SPECIAL MATRIX STRUCTURES

The following subsections explain some of the matrices that have special structures and are relevant to our discussion here.

20.2.1 Plane Rotation (Givens) Matrix

A 5 × 5 plane rotation (or Givens) matrix G_pq is one that looks like the identity matrix except for elements that lie in the locations pp, pq, qp, and qq. Such a matrix is labeled G_pq. For example, the matrix G₄₂ takes the form

(20.2)

where c = cos θ and s = sin θ. The notation commonly used is that the subscript refers to the element that has the negative sin value, which is element at row 4 and column 2 in our example.

Givens matrix is an orthogonal matrix and we have . Premultiplying a matrix A by G_pq modifies only rows p and q. All other rows are left unchanged. The elements in rows p and q become

(20.3)

(20.4)

20.2.2 Banded Matrix

A banded matrix with lower bandwidth p and upper bandwidth q implies that all its nonzero elements lie in the main diagonal, the lower p subdiagonals and the upper q superdiagonals. All other elements are zero, that is, when i > j + p and j > i + q. In that case, matrix A will have nonzero p subdiagonal elements and nonzero q superdiagonal elements. An example of a banded matrix with lower bandwidth p = 2 and upper bandwidth q = 3 has the following structure where × denotes a nonzero element:

(20.5)

20.2.3 Diagonal Matrix

A diagonal matrix D is a special case of a banded matrix when p = q = 0 and only the main diagonal is nonzero. A 5 × 5 diagonal matrix D is given by

(20.6)

We can write the above diagonal matrix in a condensed form as

(20.7)

where d_i = d_ii.

20.2.4 Upper Triangular Matrix

An upper triangular matrix U is a special case of a banded matrix when p = 0 and only the main diagonal and the first q superdiagonals are nonzero.

20.2.5 Lower Triangular Matrix

A lower triangular matrix L is a special case of a banded matrix when q = 0 and only the main diagonal and the first p subdiagonals are nonzero.

20.2.6 Tridiagonal Matrix

A tridiagonal matrix is a special case of a banded matrix when p = q = 1 and only the main diagonal, the first superdiagonal, and first subdiagonal are nonzero. A 5 × 5 tridiagonal matrix A is given by

(20.8)

20.2.7 Upper Hessenberg Matrix

An n × n upper Hessenberg matrix is a special case of a banded matrix when p = 1 and q = n and the elements of the diagonal, the superdiagonals, and the first subdiagonal are nonzero. An upper Hessenberg matrix has h_ij = 0 whenever j < i − 1. A 5 × 5 upper Hessenberg matrix H is given by

(20.9)

20.2.8 Lower Hessenberg Matrix

A lower Hessenberg matrix is the transpose of an upper Hessenberg matrix.