The Matrix as Operator

An operator is a mathematical instruction. For example, the operator d/dx is the instruction to differentiate once with respect to a . Matrices in general, and the matrix R of Chapter 6 in particular, are operators. The matrix R is an instruction to rotate a part of the operand matrix through a certain angle, 0 as in Eq. (6-62). 

The product of matrix operators is an operator. For example, rotation through 90°, followed by another rotation through 90° in the same direction and in the same plane, is the same as one rotation through 180° 

Thus the Jacobi procedure, by making many rotations of the elements of the operand matrix, ultimately arrives at the operator matrix that diagonalizes it. Mathematically, we can imagine one operator matr ix that would have diagonalized the operand matr ix R, all in one step 

The matrix A in Eq. (7-21) is comprised of orthogonal vectors. Orthogonal vectors have a dot product of zero. The mutually perpendicular (and independent) Cartesian coordinates of 3-space are orthogonal. An orthogonal n x n such as matr ix A may be thought of as n columns of n-element vectors that are mutually perpendicular in an n-dimensional vector space. 

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