R07 - ENEE 241 01 * READING ASSIGNMENT 7 Tue 02/24 Lecture...

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Unformatted text preview: ENEE 241 01 * READING ASSIGNMENT 7 Tue 02/24 Lecture Topic: introduction to matrix inversion; Gaussian elimination Textbook References: sections 2.4, 2.5, 2.6 Key Points: • A n × n matrix A is nonsingular, i.e., it possesses an inverse A- 1 , if and only if the equation Ax = b has a unique solution x (equal to A- 1 b ) for every b . • AA- 1 = I • ( A T )- 1 = ( A- 1 ) T • ( AB )- 1 = B- 1 A- 1 • A triangular matrix is nonsingular if all entries on its leading diagonal are nonzero. It can be inverted using forward or backward substitution. • In Gaussian elimination, the system of simultaneous equations Ax = b is reduced to an upper triangular system Ux = c , which is then solved using backward substitution. • The process of obtaining U from A (and c from b ) is known as forward elimination. Theory and Examples: 1 . Given the output of a linear system for which the matrix A is known, is it possible to determine its input? This would involve solving the equation Ax = b where x is the unknown input vector of dimension n , and y = b is the observed output vector of dimension m . 2 . If the entries of A are random real numbers (with infinite precision), the following statements can be made about Ax = b in probabilistic terms: • If m < n , any vector b is almost certainly a valid system output, in which case a solution x exists. The solution is not, however unique, and thus the “true” input vector cannot be determined. • If m > n , a randomly chosen vector b is almost certainly not a valid system output, in which case no solution x exists....
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This note was uploaded on 10/18/2011 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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R07 - ENEE 241 01 * READING ASSIGNMENT 7 Tue 02/24 Lecture...

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