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Transcript
Determinants and Matrices 行列式與矩陣 Chapters 17 & 18 Gialih Lin, Ph. D. Professor 17.1 Concepts • Linear algebra :matrix • The concept of determinants has its origin in the solution of simultaneous linear equations. 聯立方程式 • a1x+b1y=c1 • a2x+b2y=c2 • x=(c1b1-b1c2)/(a1b2-b1a2) • y=(a1c2-c1a2)/(a1b2-b1a2) determinant • The solution of the system can be written in the form • x=D1/D D= D1= • y=D2/D D2= a1 b1 a2 b2 c1 b1 c2 b2 a1 c1 a2 c2 17.2 Determinants of order 3 • a1x+b1y+c1z=d1 a2x+b2y+c2z=d2 a3x+b3y+c3z=d3 • x=D1/D • y=D2/D • z=D3/D • Gramer’s rule D 3= a1 b1 d1 a2 b2 d2 a3 b3 d3 a1 D= b1 c1 a2 b2 c2 b3 c3 a3 D1= D2= d1 b1 c1 d2 b2 c2 d3 b3 c3 a1 d1 c1 a2 d2 c2 a3 d3 c3 Simultaneous equations, l=1 a1 a2 a3 a1 a2 a3 a1 a2 a3 b1 b2 b3 b1 b2 b3 b1 b2 b3 x y z c1 c2 c3 x y z c1 c2 c3 c1 c2 c3 x y z - l = - 1 0 0 0 1 0 l 0 0 1 d1 d2 d3 d1 d2 d3 d1 d2 d3 =0 =0 Hermitian transformation • Change the length of vector r but not its direction • Hr = h r r s • h is a real number H11 H12 H13 H21 H22 H23 H31 H32 H33 a1 a2 a3 = h a1 a2 a3 17.3 The general case Wave functions y1 = a11f1+a12f2+a13f3+ ….+a1nfn y2 = a21f1+a22f2+a23f3+ ….+a2nfn • .. ….. yn = an1f1+an2f2+an3f3+ ….+annfn Quantum mechanics • Any function, y, can be expressed in terms of its components for a set of basis functions, fi, of unit length (column matrices or column vectors, see Chapter 18) y= a1f1+a2f2+a3f3+ ….+anfn F= b1f1+b2f2+b3f3+ ….+bnfn a1 a2 . . . an 17.4 The solution of linear equations • Gramer’s rule • D=0 • no solution exists because the equations are inconsistent. The equations are said to be linearly dependent, and each equation can be expressed as a linear combination of the others. Secular equations • • • • Shrödinger equation Hy = Ey In the form of matrix H is an nxn matrix for an n order wave function (y) • E is a constant called as eigenvalue (See chapter 18) Hermitian transformation and eigenvalue (see Chapter 19) • Change the norm of function y, but keep the relative magnititudes of its components along its basis functions the same. (see Section 18.5 linear transformation) • Hy = h F • h is a real number, and y is said to be an eigenfunction of operator H. (see chapter 19 The matrix eigenvalue problem) The operator H is said to be Hermitian when (as in quantum mechanics) the eigenvalue h is real. H11 H12 ... H1 n H21 H22 ... H2n . . . Hn1 Hn2 ... Hnn a1 a2 . . . an = h a1 a2 . . . an Secular determinant • The eigenvalue, h, of the Hermitian operator, H, are calculated by solving the secular determinant (H11 -h) H12 ... H21 (H22-h) ... . . . Hn1 Hn2 ... H 1n H 2n (Hnn-h) =0 18 Matrices and linear transformations • • • • • • 18.1 concepts 3x3 (square) matrix Row Column Trace Tr A = a1+b2+c3 A= a1 a2 a3 b1 b2 b3 c1 c2 c3 Vectors • A matrix containing a single column only is called a column matrix or column vector; a matrix containing one row only is a row matrix or row vector. The elements of a vector are called components. Quantum mechanics • Any function, y, can be expressed in terms of its components for a set of basis functions, fi, of unit length (column matrices or column vectors, see Chapter 18) y= a1f1+a2f2+a3f3+ ….+anfn F= b1f1+b2f2+b3f3+ ….+bnfn a1 a2 . . . an Matrix multiplication • C=AB • The number of columns of A=the number of rows of B. if A is mxn and B is nxp, the procdut C is mxp Hermitian transformation • Change the length of vector r but not its direction • Hr = h r r s • h is a real number H11 H12 H13 H21 H22 H23 H31 H32 H33 a1 a2 a3 = h a1 a2 a3 Multiplication by a unit matrix • If A is an mxn matrix and Im and In are the unit matrices of orders m and n, respectively, then • Im A= A =A In Simultaneous equations, l=1 a1 a2 a3 a1 a2 a3 a1 a2 a3 b1 b2 b3 b1 b2 b3 b1 b2 b3 x y z c1 c2 c3 x y z c1 c2 c3 c1 c2 c3 x y z - l = - 1 0 0 0 1 0 l 0 0 1 d1 d2 d3 d1 d2 d3 d1 d2 d3 =0 =0 18.5 Linear transformations • • • • x’ = Ax Simultaneous transformations Consecutive transformations Inverse transformations Hermitian transformation • Change the length of vector r but not its direction • Hr = h r r s • h is a real number H11 H12 H13 H21 H22 H23 H31 H32 H33 a1 a2 a3 = h a1 a2 a3