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mtxmthds_1982

# mtxmthds_1982 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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parenleftBigg parenrightBigg parenleftBigg parenrightBigg parenleftBigg parenrightBigg MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Revised February, 1982 NOTES ON MATRIX METHODS 1. Matrix Algebra Margenau and Murphy, The Mathematics of Physics and Chemistry, Chapter 10, give almost all the back- ground relevant for physical applications. Schi ff , Quantum Mechanics, Chapter VI, and many other texts, give useful brief summaries. Here we shall note only a few of the most important definitions and theorems. The matrix is a square or rectangular array of numbers that can be added to or multiplied into another matrix according to certain rules. For a matrix A , the elements are denoted A ij , where indices refer to rows and columns respectively. Addition . If two matrices A and B have the same rank (same number of rows and the same number of columns) they may be added element by element. If the sum matrix is called C = A + B then C i j = A i j + B i j . (1.1) The addition is commutative: A + B = B + A . (1.2) Multiplication . Two matrices A and B can be multiplied together to give C = AB by the following rule C i j = Σ k A ik B k j . (1.3) If A has m rows and n columns, B must have n rows and m columns if C is a square matrix with m rows and m columns. Each element C i j of the product is a sum of products along the i th row of A and the j th column of B . For example, for 2 × 2 matrices: A 11 A 12 B 11 B 12 A 11 B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 = A 21 A 22 B 21 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 Note that in general matrix multiplication is not commutative: AB BA . (1.4) From the definitions it follows at once that the distributive law of multiplication still holds for matrices: A ( B + C ) = AB + AC . (1.5)
vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle Handout: Notes on Matrix Methods Revised February, 1982 Page 2 Likewise, the associative law holds: A ( BC ) = ( AB ) C . (1.6) Inverse of a matrix . Matrix division is not defined. A matrix may or may not possess an inverse A 1 , which is defined by AA 1 = E and A 1 A = E (1.7) where E is the unit matrix (consisting of 1’s along the diagonal and 0’s elsewhere). The Matrix A is said to be nonsingular if it possesses an inverse and singular if it does not. If A is nonsingular and of finite rank, it can be shown to be square and the i j element of its inverse is just the cofactor of A ji divided by the determinant of A . Thus A is singular if its determinant vanishes. It is readily verified from (1.3) and (1.7) that for nonsingular matrices ( ABC ) 1 = C 1 B 1 A 1 . (1.8) The determinant

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mtxmthds_1982 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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