Complex Inner Product Spaces Part 1

Published 10/4/2025

Unitary Matrices

Computational Problems

Problem 1.1

\text{Let }r_1=(1+i,\,2)\in\mathbb{C}^2.\ \text{Build a }2\times2\text{ unitary }U\text{ with }r_1\text{ as its first row by:}\\[4pt] \text{(a) Normalizing }r_1.\quad \text{(b) Finding all }r_2=(x,y)\text{ with }\overline{r_1}\cdot r_2=\bar a x+\bar b y=0.\quad \text{(c) Parametrizing }r_2\text{ via a phase }e^{i\theta}.\\[10pt] \text{Detailed explanation:}\\ \text{(a) } \|r_1\|=\sqrt{|1+i|^2+|2|^2}=\sqrt{2+4}=\sqrt{6},\ \ \widehat r_1=\frac{1}{\sqrt{6}}\,(1+i,\,2).\\[6pt] \text{(b) Solve }\bar a x+\bar b y=0\text{ with }a=\tfrac{1+i}{\sqrt{6}},\ b=\tfrac{2}{\sqrt{6}}.\ \text{A spanning solution is }(-\bar b,\,\bar a).\\ \text{Indeed, } \bar a(-\bar b)+\bar b(\bar a)=-\bar a\bar b+\bar b\bar a=0.\ \text{To make }r_2\text{ unit, take any phase }e^{i\theta}\text{ and set }\\ r_2=e^{i\theta}\,(-\bar b,\,\bar a)=e^{i\theta}\!\left(-\tfrac{2}{\sqrt{6}},\ \tfrac{1-i}{\sqrt{6}}\right),\ \text{which has }\|r_2\|=1\ \text{and }\overline{\widehat r_1}\cdot r_2=0.\\[6pt] \text{(c) Hence all }2\times2\text{ unitaries with first row }\widehat r_1\text{ are }\\ U(\theta)=\begin{bmatrix} \tfrac{1+i}{\sqrt{6}} & \tfrac{2}{\sqrt{6}}\\[6pt] -\tfrac{2}{\sqrt{6}}e^{i\theta} & \tfrac{1-i}{\sqrt{6}}e^{i\theta} \end{bmatrix},\ \ \theta\in\mathbb{R}.\\[8pt] \text{Answer: } \widehat r_1=\big(\tfrac{1+i}{\sqrt{6}},\ \tfrac{2}{\sqrt{6}}\big),\ r_2=e^{i\theta}\!\left(-\tfrac{2}{\sqrt{6}},\ \tfrac{1-i}{\sqrt{6}}\right),\ U(\theta)=\begin{bmatrix} \tfrac{1+i}{\sqrt{6}} & \tfrac{2}{\sqrt{6}}\\ -\tfrac{2}{\sqrt{6}}e^{i\theta} & \tfrac{1-i}{\sqrt{6}}e^{i\theta} \end{bmatrix}.

Problem 1.2

\text{Let }r_1=(3,\,-i).\ \text{(a) Normalize }r_1.\ \text{(b) Find all }r_2=(x,y)\ \text{with}\ \bar a x+\bar b y=0.\ \text{(c) Parametrize with }e^{i\theta}.\\[6pt] \textbf{Answer: } \|r_1\|=\sqrt{10},\ \widehat r_1=\big(\tfrac{3}{\sqrt{10}},\ -\tfrac{i}{\sqrt{10}}\big).\ \text{ All }r_2=e^{i\theta}\,(-\bar b,\,\bar a)=e^{i\theta}\!\left(\tfrac{i}{\sqrt{10}},\ \tfrac{3}{\sqrt{10}}\right).\\ U(\theta)=\begin{bmatrix} \tfrac{3}{\sqrt{10}} & -\tfrac{i}{\sqrt{10}}\\[6pt] \tfrac{i}{\sqrt{10}}e^{i\theta} & \tfrac{3}{\sqrt{10}}e^{i\theta} \end{bmatrix},\ \theta\in\mathbb{R}.

Problem 1.3

\text{Let }r_1=(2-i,\,1+2i).\ \text{(a) Normalize }r_1.\ \text{(b) Find all }r_2\text{ with }\bar a x+\bar b y=0.\ \text{(c) Parametrize with }e^{i\theta}.\\[6pt] \textbf{Answer: } |2-i|^2=5,\ |1+2i|^2=5,\ \|r_1\|=\sqrt{10},\ \widehat r_1=\big(\tfrac{2-i}{\sqrt{10}},\ \tfrac{1+2i}{\sqrt{10}}\big).\\ r_2=e^{i\theta}\,(-\bar b,\,\bar a)=e^{i\theta}\!\left(-\tfrac{1-2i}{\sqrt{10}},\ \tfrac{2+i}{\sqrt{10}}\right),\\ U(\theta)=\begin{bmatrix} \tfrac{2-i}{\sqrt{10}} & \tfrac{1+2i}{\sqrt{10}}\\[6pt] -\tfrac{1-2i}{\sqrt{10}}e^{i\theta} & \tfrac{2+i}{\sqrt{10}}e^{i\theta} \end{bmatrix},\ \theta\in\mathbb{R}.

Problem 1.4

\text{Let }(a,b)\neq(0,0).\ \text{(a) Normalize }r_1=(a,b).\ \text{(b) Solve }\bar a x+\bar b y=0\text{ for unit }r_2.\ \text{(c) Parametrize all }U.\\[6pt] \textbf{Answer: } \widehat r_1=\dfrac{1}{\sqrt{|a|^2+|b|^2}}\,(a,b).\ \text{ All orthogonal unit rows are }r_2=e^{i\theta}\,(-\bar b,\ \bar a)\ /\ \sqrt{|a|^2+|b|^2}.\\ \text{Thus } U(\theta)=\dfrac{1}{\sqrt{|a|^2+|b|^2}} \begin{bmatrix} a & b\\[4pt] -\bar b\,e^{i\theta} & \bar a\,e^{i\theta} \end{bmatrix},\ \theta\in\mathbb{R}.

Problem 1.5

\text{Let a desired first row be }r_1=(\cos\phi,\ e^{i\psi}\sin\phi),\ \phi\in[0,\pi/2],\ \psi\in\mathbb{R}.\\ \text{(a) Verify it is unit. (b) Find all }r_2\text{ with }\bar a x+\bar b y=0.\ (c)\ \text{Parametrize }U.\\[6pt] \textbf{Answer: } \|r_1\|^2=\cos^2\phi+\sin^2\phi=1.\ r_2=e^{i\theta}\,(-\bar b,\ \bar a)=e^{i\theta}\,(-e^{-i\psi}\sin\phi,\ \cos\phi).\\ U(\theta)=\begin{bmatrix} \cos\phi & e^{i\psi}\sin\phi\\[6pt] -\,e^{-i\psi}\sin\phi\ e^{i\theta} & \cos\phi\ e^{i\theta} \end{bmatrix},\ \theta\in\mathbb{R}.

Complex Inner Product Spaces Part 1

Unitary Matrices

Computational Problems

Problem 1.1

Problem 1.2

Problem 1.3

Problem 1.4

Problem 1.5

Table of Contents