Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
3.18 If A ~ ~B ˆ 0, show that one of these matrices must have zero determinant.
3.19 Given the Pauli spin matrices (which are very important in quantum
mechanics)
1 ˆ 0 1
1 0
; 2 ˆ 0 i
i 0
; 3 ˆ 1 0
0 1
(note that the subscripts x; y, and z are sometimes used instead of 1, 2, and
3). Show that
(a) they are hermitian,
(b) 2 i ˆ ~I; i ˆ 1; 2; 3
(c) as a result of (a) and (b) they are also unitary, and
(d) [ 1 ; 2 Šˆ2I 3 et cycl.
Find the inverses of 1 ; 2 ; 3 :
3.20 Use a rotation matrix to show that
sin… 1 ‡ 2 †ˆsin 1 cos 2 ‡ sin 2 cos 1 :
3.21 Show that: Tr ~A ~B ˆ Tr ~B ~A and Tr ~A ~B ~C ˆ Tr ~B ~C ~A ˆ Tr ~C ~A ~B:
3.22 Show that: (a) the trace and (b) the commutation relation between two
matrices are invariant under similarity transformations.
3.23 Determine the eigenvalues and eigenvectors of the matrix
a b
~A ˆ :
b a
;
Given
0
5 7
1
5
B ~A ˆ @ 0 4
C
1 A;
2 8 3
®nd a matrix ~S that diagonalizes ~A, and show that ~S 1 ~A ~S is diagonal.
3.25 If ~A and ~B are square matrices of the same order, then
det( A ~ ~B† ˆdet… A† ~ det… ~B†: Verify this theorem if
~A ˆ 2 1
3 2
; ~B ˆ
7 2
3 4
3.26 Find a common set of eigenvectors for the two matrices
0 p
p
1 0 p
p
1
1 6 2
10 6 2
B p
p
C B p
p
C
~A ˆ @ 6 0 3 A; ~B ˆ @ 6 9 3
p
p
p
2 3 2
A:
p
2 3 11
3.27 Show that two hermitian matrices can be made diagonal if and only if they
commute.
142
: