_TZ_6393-periastron.dvi
MATH 3221 ADVANCED LINEAR ALGEBRA
DAILY ASSIGNMENT MARCH 29, 2023
41. Let T : C2 → C2 be the operators whose matrix with respect to the standard basis is
[
1 i
i 1
]
.
Show that T is normal (though it is neither self-adjoint, skew-adoint, or unitary), and find
an orthonormal basis of C2 of eigenvectors of T .
42. Here is one characterisation of normal operators: T is normal iff T = T1 + iT2, where T1
and T2 are self-adjoint operators that commute. Show this as follows.
Let V be a complex inner-product space, and T a linear operator on V . Define
T1 =
1
2
(
T + T
∗
)
and T1 =
1
2i
(
T − T
∗
)
.
(a) Show that T1 and T2 are self-adjoint and that T = T1 + iT2.
(Note the analogy to the real and imaginary parts of a complex number, if we think
of matrix adjoint as analogous to complex conjugation.)
(b) Show that this decomposition is unique — that is, show that if T = U1 + iU2 with
oth U1, U2 self-adjoint, then U1 = T1 and U2 = T2.
(c) Prove that T is normal iff T1 and T2 commute.
Thus, normal operators are ones whose “real part” and “imaginary part” commute —
ut only when we interpret “real part” and “imaginary part” properly. (In particular,
this does not mean that taking the real and imaginary parts of the individual entries of
a matrix will give matrices that commute!) I don’t know if that helps with the intuition
much, but it’s interesting!