Gerschgorins circles theorem

This theorem has important implications in the box model theory. It states that every eigenvalue of A x , possibly complex, lies in the complex plane inside at least one of the circles centered at the diagonal entry a and with a radius equal to the sum Z a0- (i j) of all the off-diagonal elements of the ith row. 

Let Uj be the component of u with the largest absolute value. Then, by the rule of matrix product 

Taking the modulus of each side and applying the rule for sums of modulus (Schwarz 

The same argument applies to AT and may be used to calculate Gershgorin s circles with respect to rows instead of columns. 

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The next result is also general and, while the conditions are not often met, it is an important tool when it can be applied. The theorem is called the Gerschgorin circle theorem. An excellent general reference on matrices is Lancaster and Tismenetsky [LT], and most of the results here are quoted from that source. Another important source is Berman and Plem-mons [BP], particularly for special results on nonnegative matrices. 

For a general Jacobian matiix pertaining to C components and N theoretical trays, as shown by Distefano [Am. Jnst. Chem. Eng. J., 14, 946 (1968)]. Gerschgorin s circle theorem (Varga, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.J., 1962) may be employed to obtain bounds on the maximum and minimum absolute eigenvalues. Accord-... 

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