By R. K. Eisenschitz Dr.Phil., D.Sc., F.Inst.P. (auth.)
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Extra resources for Matrix Algebra for Physicists
A) and the existence of determinants of every order is thus proved. 27 DETERMINANTS EXERCISES I. Evaluate the determinants -3 -2 2 3 4 2 1 1 -1 -1 1 3 2 2 and 1 0 2 1 4 3 2 3 -1 1 -3 2. Show that the matrix 3 4i 3 -1 -5i 1 has areal determinant. 3. Using the theorems of Section 11, show that the determinant of a matrix must vanish if two rows are equal to each other. 4. jm = bjm det A j=l 5. 2) convert the matrix A= to rr~ I1 1 1 -1 -1 1 ~il 0 0 I 2 0 B= 1 1 2 -2 1 0 2 and thus derive det A. ) ~l CHAPTER 5 MATRICES AND LINEAR EQUATIONS 15.
The sub-matrix derived from B by deleting the first row and column must have linearly independent columns. For if, for non-vanishing q, .. Lbikq~=O (j=2 ... n) k~2 28 MATRICES AND LINEAR EQUATIONS it would be possible to choose ql dependence L 29 0 and to establish the linear = n contrary to premises. In the present instance bjkqk = 0 k~1 det B = bllßll ßll being the cofactor to bn . B) is valid for (n - 1) x (n - 1) matrices this cofactor is non-zero. As bn :;ce 0 this would show that det B :;ce O.
THEOREM Let U be a unitary matrix and A' = U-IAU be diagonal. Then the diagonal elements of Aare a;; = 2: k U;ka;kukjl = 2: a~k IUjk 12 k and they are aeeording to premises non-negative. By a similar argument it follows that the diagonal elements of all unitary transforms of A are non-negative. The sum A + B is Hermitean. The diagonal elements of A and of DIAGONALIZATION OF MATRICES 51 Bare accordingly non-negative and if V is unitary the diagonal elements of V-lAV + V-lBV = V-l(A + B)V are also nonnegative.
Matrix Algebra for Physicists by R. K. Eisenschitz Dr.Phil., D.Sc., F.Inst.P. (auth.)