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M _ I)! e , complex A = Ct. ± uo -- e" cos tot, e" sin cot; I t e~ Ism ' cot . -t e~ cos cot I! ' 1! ' tm - l tm - l ~I t ~/' t (m _ I)! e cos w '(m _ I)! e sm w . 3) m is the multiplicity of the eigenvalue A. 3) is applied to each distinct real eigenvalue and then to each distinct complex conjugate pair. Otherwise the order is immaterial. Step 3. Determine the real n x n matrix B such that By(t) = y(t). 4) This can be done by inspection, due to the fact that the derivative of each Yi(t) assigned by the correspondence rule is a linear combination of the coordinate functions composing y(t).

Programs for solving large systems of linear algebraic equations ordinarily do not reduce the augmented matrix to the echelon form described earlier in this chapter. The more widely used method of Gauss elimination row reduces the matrix to an upper-triangular form and then completes the computation by back substitution, starting with the bottom nonzero row, rather than proceeding with back elimination to reach the echelon form. To compare the efficiency of algorithms, suppose that each division and each multiplication-addition is counted as one operation.

13) is thereby proved. There is a matrix generalization of the multiplicative identity 1 in :F. 16) IA = A = AI has a unique n x n matrix solution I over:F holding for all n x n A over :F; namely, (I)ij =. bijwhere bij takes on the value 1 in :F when i = j and the value 0 in :F when i '" j. I is called the n x n (multiplicative) identity matrix over:F and is denoted by In when its size needs to be emphasized. 18) (cA)B = A(cB) for all matrices A, B over :F of compatible size. Thus the definitions of matrix addition, multiplication, and scalar multiplication provide extensions of many of the formal rules of algebra in :F to matrices over :F.

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