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Unformatted text preview: BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Linear Algebra BME 5020 Computer and Mathematical Application in Bioengineering 2006 Jingwen Hu September 28, 2006 Gauss Elimination LU Decomposition Matrix Inverse Contents BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Introduction BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University An equation of the form ax+by+c=0 or equivalently ax+by=c is called a linear equation in x and y variables. ax+by+cz=d is a linear equation in three variables, x, y , and z . Thus, a linear equation in n variables is a 1 x 1 +a 2 x 2 + +a n x n = b A solution of such an equation consists of real numbers c 1 , c 2 , c 3 , , c n . If you need to work more than one linear equations, a system of linear equations must be solved simultaneously. BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Introduction BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University A system of equations with the form: n n nn n n n n n n b x a x a x a b x a x a x a b x a x a x a = + + + = + + + = + + + ... ... ... ... ... 2 2 1 1 2 2 2 22 1 21 1 1 2 12 1 11 is typically referred to as linear algebraic equations. Frequently, these equations are expressed as [A] {x} = {b} BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix Symmetric matrix: a ij = a ji BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 8 7 2 7 3 1 2 1 5 ] [ A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix Diagonal matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 33 22 11 ] [ a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix Identity matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 1 1 1 1 ] [ I BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix Upper triangular matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 34 33 24 23 22 14 13 12 11 ] [ a a a a a a a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix Lower triangular matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 43 42 41 33 32 31 22 21 11 ] [ a a a a a a a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University Special Matrix Banded matrix BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University = 44 43 34 33 32 23 22 21 12 11 ] [ a a a a a a a a a a A BME5020 2006, Fall Semester Biomedical Engineering, Wayne State University...
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