Applied Mathematics I Physics Department, National Taiwan University
Professor Jiun-Huei Proty Wu Semester 112-2 (Spring 2024)
Example Sheet 3
1. Consider a vector 𝒗 = (1,2,3) in 𝑅3 . Now we define a new coordinate system
where the three unit vectors (the new basis) along the three axes are 𝑒̂1′ ≡ |𝑒1′ ⟩ =
(1/√3, 1/√3, 1/√3), 𝑒̂2′ ≡ |𝑒2′ ⟩ = (−1/√3, 1/√3, 1/√3), 𝑒̂3′ ≡ |𝑒3′ ⟩ =
⟨𝑒1′ |
(0, −1/√2, 1/√2). We also define 𝑵 = (⟨𝑒2′ |). Answer the following questions.
⟨𝑒3′ |
(a) What are the new coordinates 𝒗′ of 𝒗 in this new coordinate system?
(b) Compute 𝑵𝒗. This is the righthand side of equation (2) in section 4.1 of the
lecture notes.
(c) If we write 𝒗′ = 𝑴𝒗, then what is 𝑴?
(d) Compute |𝑵| and |𝑴|.
(e) Compute Tr(𝑵) and Tr(𝑴).
(f) Is 𝑵 orthogonal? (hint: orthogonality condition)
(g) Is 𝑴 orthogonal? (hint: orthogonality condition)
(h) What is |𝒗|?
(i) What is |𝒗′|?
2. For the same 𝒗 = (1,2,3) in 𝑅3 , consider another new coordinate system where
the three unit vectors (the new basis) along the three new axes are 𝑒̂1′ = (𝑎, 𝑏, 𝑐 ),
1 1
𝑒̂2′ = (−𝑎, 𝑏, 𝑐 ), 𝑒̂3′ = (0, − , ), and 𝑎, 𝑏, 𝑐 > 0. We define 𝑵 in the same
√2 √2
way, and we already know that 𝑵 is orthogonal. Find 𝑎, 𝑏, 𝑐.
3. Construction of orthonormal basis (Gram–Schmidt process)
This is an exercise to construct a set of orthonormal basis {|𝒆𝒊 ⟩} (i=1…N) from a
given set of non-orthogonal basis {|𝒙𝒊 ⟩} in C 𝑵 . As instructed during the lectures,
the process is the following:
(i) Normalise the first vector to obtain |𝒆𝟏 ⟩ = |𝒙𝟏 ⟩/||𝒙𝟏 ⟩|.
(ii) Remove the projected components of the previous |𝒆𝒋 ⟩ (j=1… i-1) from
|𝒙𝒊 ⟩:
|𝒙𝒊 ′⟩ = |𝒙𝒊 ⟩ − ⟨𝒆𝒋 |𝒙𝒊 ⟩|𝒆𝒋 ⟩ (Einstein summation has been used here.)
(iii) Normalise this result to obtain |𝒆𝒊 ⟩:
|𝒆𝒊 ⟩ = |𝒙𝒊 ′⟩/||𝒙𝒊 ′⟩|
(iv) Repeat steps (ii) and (iii) until i = N.
Here are some practical exercises for you:
(a) Consider a vector space in R𝟒 . Given a set of basis |𝒙𝟏 ⟩ = (1, 1, 0, 0), |𝒙𝟐 ⟩ =
(0, 1, 1, 0), |𝒙𝟑 ⟩ = (0, 0, 1, 1), |𝒙𝟒 ⟩ = (0, 0, 0, 1), use the above method to
find a set of orthonormal basis, where |𝒆𝟏 ⟩ = (1/√2, 1/√2, 0, 0)
(b) Consider a vector space in C 𝟐 . Given a set of basis |𝒙𝟏 ⟩ = (1, −𝑖 ), |𝒙𝟐 ⟩ =
(𝑖, 𝑖 ), use the above method to find a set of orthonormal basis.
(c) For a vector space in C 𝑵 , in step (ii) of the above method, can we choose
|𝒙𝒊 ′⟩ = |𝒙𝒊 ⟩ − ⟨𝒙𝒊 |𝒆𝒋 ⟩|𝒆𝒋 ⟩? And why? (hint: multiply both sides with ⟨𝒆𝑘 |
where 𝑘 < 𝑖.)
4. Transformation of coordinate system
Consider an 𝑅3 vector space. A vector defined with an orthonormal basis is 𝒗 =
(1, 1, 0). Its new coordinates in the transformed space (coordinate system) is 𝒗′ =
𝑴𝒗 = (1, −1, 0), where 𝑴 is the transformation matrix for the coordinate
system. Answer the following questions.
(a) If the transformation of coordinate system was done through the Euler matrix,
i.e. 𝑴 = 𝑹𝑧 (𝛾)𝑹𝑦 (𝛽)𝑹𝑧 (𝛼), where 𝛼, 𝛾 ∈ [−𝜋, 𝜋] and 𝛽 ∈ [0, 𝜋], how
many possible sets of (𝛼, 𝛽, 𝛾) are there?
(b) Following (a), if 𝛼 = 0, what are the two possible sets of (𝛽, 𝛾)? And what
are the matrices 𝑴 corresponding to these two cases?
(c) If the transformation was done through two successive reflections of 𝑶𝟏 and
𝑶𝟐 , i.e. 𝑴 = 𝑶𝟐 𝑶𝟏 , and 𝑶𝟏 has a unit normal of (0, 𝑏, 𝑐) where 𝑏, 𝑐 ∈ 𝑅,
find 𝑶𝟐 .
5. Consider the following four matrices:
1 0 1 −𝑖 𝑖 1
𝐀=( ),𝑩 = ( ),𝑪 = ( ) , 𝑫 = 𝑩 ⊗ 𝑪.
0 1 𝑖 𝑖 1 1
(a) Which of these are Hermitian?
(b) Which of these are unitary?
(c) Which of these are normal?
1 −1 𝑔11 𝑔12
6. Given 𝑭 = ( ), 𝑮 = (𝑔 𝑔22 ), Tr(𝑮) = 2, G is Hermitian, and
−3 1 21
[𝑭, 𝑮] = 𝟎, find G.
𝑎 𝑑 𝑒 𝑗 𝑚 𝑛
7. Given 𝑨 = [𝑑 ∗ 𝑏 𝑓 ] and 𝑩 = [𝑚∗ 𝑘 𝑜 ], where 𝑎, 𝑏, 𝑐, 𝑗, 𝑘, 𝑙 ∈ 𝑅 and
𝑒∗ 𝑓∗ 𝑐 𝑛∗ 𝑜∗ 𝑙
are all different. All the other components are complex numbers.
(a) If 𝑑 = 𝑒 = 𝑓 = 0, and 𝑨 and 𝑩 commute (i.e. [𝑨, 𝑩] = 𝟎, or 𝑨𝑩 = 𝑩𝑨),
find m and n.
𝑝𝑖 3 4𝑖
(b) If [𝑨, 𝑩] = ( 𝑠 𝑞 + 2𝑖 5 ) where 𝑝, 𝑞, 𝑟 ∈ 𝑅, find p and s .
𝑡 𝑢 𝑟+𝑖
(c) If 𝑨 and 𝑩 are both unitary, 𝑨 ≠ 𝑩, and they anti-commute (i.e. 𝑨𝑩 =
−𝑩𝑨), find (𝑎 + 𝑏 + 𝑐) and (𝑗 + 𝑘 + 𝑙).
