Inversion and Diagonalization

Looking at the matrix equation Ax = b, one would be tempted to divide both sides by matrix A to obtain the solution set x = b/A. Unfortunately, division by a matrix is not defined, but for some matrices, including nonsingular coefficient matrices, the inverse of A is defined. 

The unit matrix, I, with an = 1 and Gy = 0 for i plays the same role in matrix algebra that the number 1 plays in ordinary algebra. In ordinary algebra, we can perform an operation on any number, say 5, to reduce it to 1 (divide by 5). If we do the same operation on 1, we obtain the inverse of 5, namely, 1/5. Analogously, in matrix algebra, if we cany out a series of operations on A to reduce it to the unit matrix and cany out the same series of operations on the unit matrix itself, we obtain the inverse of the original matrix A .  

The attractive feature in matrix inversion is seen by premultiplying both sides of Ax = b by A ,  

This means that onee A is known, it ean be multiplied into several b veetors to generate a solution set x = A b for each b vector. It is easier and faster to multiply a matrix into a vector than it is to solve a set of simultaneous equations over and over for the same coefficient matrix but different b vectors. 

Plijcc the orij ini)l eoeffieient matrix next to the unit matrix. Divide row 1 by 2 ami subtraet the result from row 2, This is a linear operation, and it does not ehange the solution set. 

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