By S. Twomey

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**Additional resources for Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements**

**Sample text**

000001 This implies the contradiction 0 = 0 . 0 0 0 0 0 1 . Finally if both small decimal fractions are omited: x + y =2 2x + 2y = 4 32 The second equation of this pair merely repeats the same information as the first and there are an infinity of solutions; any pair of values for x and y such that x + y = 2 satisfies the system of equations. It now becomes apparent that the simple nature of the problem becomes quite illustory when the system of equations are not exact statements of absolute fact.

A scalar u*v. If on the other hand, a column matrix v, an (n X 1) matrix, premultiplies a row vector u*, a (1 X n) matrix, the result — the "outer product" — is an n X n matrix vu*. It contains in its (/, j) element the product of the z'th ele ment of v with the jth element of u. 3 INVERSE OF A MATRIX Premultiplication of a vector x by a matrix A gives a new vector, y. The question immediately arises whether a matrix exists which premultiplying y gives x. It is always possible to concoct such a matrix, but if the relationship Ax = y, By = x holds for all x andy, B is a matrix which inverts the effect of A.

42 the premultiplier is a row vector u* and the postmultiplier a column vector v. , a scalar u*v. If on the other hand, a column matrix v, an (n X 1) matrix, premultiplies a row vector u*, a (1 X n) matrix, the result — the "outer product" — is an n X n matrix vu*. It contains in its (/, j) element the product of the z'th ele ment of v with the jth element of u. 3 INVERSE OF A MATRIX Premultiplication of a vector x by a matrix A gives a new vector, y. The question immediately arises whether a matrix exists which premultiplying y gives x.