Chapter3_17 - means that its determinant is zero The...

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PHY4604 R. D. Field Department of Physics Chapter3_17.doc University of Florida Matrices as Operators (2) The Inverse Matrix: The inverse of a ( square ) matrix is defined by I T T T T = = ~ ~ ~ ~ 1 1 where I is the identity matrix. A matrix has an inverse only if its determinant det(T) is non-zero since ) ~ det( ~ ~ 1 T C T = , where C ~ is the matrix of cofactors (the cofactor of element T ij is (-1) i+j times the determinant of the submatrix obtained from T by removing the i th row and the j th column). Note that 1 1 1 ~ ~ ) ~ ~ ( = A B B A . The Eigenvalue Equation: The eigenvalue equation is > >= v v T | | ~ λ for nonzero |v> which can be written as 0 | ) ~ ( >= v I T , where 0 is the zero matrix. Now if ) ~ ( I T has an inverse then we can multiply both sides by 1 ) ~ ( I T , and conclude that |v> = 0. But the assumption is that |v> is not zero and hence ) ~ ( I T must be singular, which
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Unformatted text preview: means that its determinant is zero. The eigenvalues are then solutions of the equation ) ~ det( 2 1 2 22 21 1 12 11 = − − − = − nn n n n n T T T T T T T T T I T L M M M M L L . Example: The eigenvalues of the 2×2 matrix − 2 1 2 1 come from solving the equation 2 1 2 1 = − − − . Thus, ) )( ( 2 1 2 1 = − − − and two eigenvalues are λ = +1/2 and λ = -1/2. Example: The eigenvalues of the 3×3 matrix − i i come from solving the equation = − − − − i i . Thus, λ ( λ 2-1)=0 and the three eigenvalues are λ = +1, λ = 0, and λ = -1....
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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