Lab 11. Complex Inner Products, Unitary Matrices, Hermitian Matrices, and Schur Decomposition: Independent Study

This lab accompanies Chapter 11: Complex Inner Products, Unitary Geometry, Hermitian Matrices, and Schur Decomposition.

The goal is to connect complex geometry with computation:

Complex inner products require conjugation.
Unitary matrices preserve length and angle.
Hermitian matrices are the complex analogue of real symmetric matrices.
Schur decomposition triangularizes every complex square matrix using a unitary basis.
Normal matrices are exactly the matrices that can be unitarily diagonalized.

This is an independent-study lab. Each main question includes a worked solution and a similar practice question.

Python practice notebook

You may use the Jupyter notebook version for longer Python practice:

Interactive lab

Study guide and worked questions

Question 1. Complex inner product

Let \[ u=\begin{bmatrix}1+i\\2\end{bmatrix}, \qquad v=\begin{bmatrix}i\\1-i\end{bmatrix}. \] Using the convention $\langle u,v\rangle=v^*u$, compute $\langle u,v\rangle$, $\langle v,u\rangle$, and $\|u\|$.

Solution

\[ \langle u,v\rangle =\overline{i}(1+i)+\overline{(1-i)}(2) =(-i)(1+i)+(1+i)2. \] Since \[ (-i)(1+i)=1-i, \qquad 2(1+i)=2+2i, \] we get \[ \langle u,v\rangle=3+i. \] Therefore \[ \langle v,u\rangle=3-i, \] and \[ \|u\|=\sqrt{|1+i|^2+|2|^2}=\sqrt6. \]

Similar practice

Let \[ a=\begin{bmatrix}2-i\\1+i\end{bmatrix}, \qquad b=\begin{bmatrix}1+i\\-i\end{bmatrix}. \] Compute $\langle a,b\rangle$, $\langle b,a\rangle$, and $\|a\|$.

Answer

\[ \langle a,b\rangle=\overline{(1+i)}(2-i)+\overline{(-i)}(1+i) =(1-i)(2-i)+i(1+i). \] Now \[ (1-i)(2-i)=1-3i, \qquad i(1+i)=i-1. \] Hence \[ \langle a,b\rangle=-2i, \qquad \langle b,a\rangle=2i. \] Also \[ \|a\|=\sqrt{|2-i|^2+|1+i|^2}=\sqrt{5+2}=\sqrt7. \]

Question 2. Projection onto a complex line

Let \[ u=\frac{1}{\sqrt2}\begin{bmatrix}1\\ i\end{bmatrix}, \qquad y=\begin{bmatrix}2\\1+i\end{bmatrix}. \] Compute the projection of $y$ onto $\operatorname{span}(u)$.

Solution

Since $u$ is unit length, \[ \operatorname{proj}_{\operatorname{span}(u)}(y)=\langle y,u\rangle u. \] Using $\langle y,u\rangle=u^*y$, \[ \langle y,u\rangle=\frac{1}{\sqrt2}\begin{bmatrix}1&-i\end{bmatrix} \begin{bmatrix}2\\1+i\end{bmatrix} =\frac{3-i}{\sqrt2}. \] Thus \[ \operatorname{proj}(y)=\frac{3-i}{2}\begin{bmatrix}1\\ i\end{bmatrix}. \]

Similar practice

Let \[ w=\frac{1}{\sqrt5}\begin{bmatrix}1+2i\\0\end{bmatrix}, \qquad z=\begin{bmatrix}3-i\\2\end{bmatrix}. \] Compute $\operatorname{proj}_{\operatorname{span}(w)}(z)$.

Answer

Because $w$ is unit length, \[ \operatorname{proj}(z)=\langle z,w\rangle w. \] Now \[ \langle z,w\rangle=w^*z =\frac{1}{\sqrt5}\overline{(1+2i)}(3-i) =\frac{1}{\sqrt5}(1-2i)(3-i) =\frac{1-7i}{\sqrt5}. \] Therefore \[ \operatorname{proj}(z)=\frac{1-7i}{5} \begin{bmatrix}1+2i\\0\end{bmatrix}. \]

Question 3. Hermitian and unitary checks

Let \[ A=\begin{bmatrix}2&1+i\\1-i&3\end{bmatrix}, \qquad U=\frac1{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}. \] Check whether $A$ is Hermitian and whether $U$ is unitary.

Solution

\[ A^*=\begin{bmatrix}2&1+i\\1-i&3\end{bmatrix}=A, \] so $A$ is Hermitian.

Also, \[ U^*U=I, \] so $U$ is unitary. Since this matrix is real, $U^*=U^T$.

Similar practice

Check whether \[ B=\begin{bmatrix}1&i\\i&1\end{bmatrix} \] is Hermitian.

Answer

\[ B^*=\begin{bmatrix}1&-i\\-i&1\end{bmatrix}\ne B. \] So $B$ is not Hermitian.

Question 4. Schur form and eigenvalues

Suppose \[ A=UTU^*, \qquad T=\begin{bmatrix} 2&5&1\\0&3&4\\0&0&-1 \end{bmatrix}. \] What are the eigenvalues of $A$?

Solution

Unitary similarity preserves eigenvalues, and the eigenvalues of an upper triangular matrix are the diagonal entries. Therefore the eigenvalues of $A$ are \[ 2,\quad 3,\quad -1. \]

Similar practice

If \[ S=\begin{bmatrix} i&2\\0&1-i \end{bmatrix} \] is a Schur form of a matrix $B$, what are the eigenvalues of $B$?

Answer

The eigenvalues are \[ i,\qquad 1-i. \]

Question 5. Normal but not Hermitian

Give an example of a normal matrix that is not Hermitian.

Solution

Let \[ D=\begin{bmatrix}i&0\\0&1\end{bmatrix}. \] Since $D$ is diagonal, it is normal: \[ D^*D=DD^*. \] But \[ D^*=\begin{bmatrix}-i&0\\0&1\end{bmatrix}\ne D, \] so $D$ is not Hermitian.

Similar practice

Show that \[ R=\begin{bmatrix}0&-1\\1&0\end{bmatrix} \] is normal but not Hermitian.

Answer

Since $R$ is real orthogonal, $R^TR=RR^T=I$, so it is normal. But \[ R^T=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\ne R, \] so it is not Hermitian.

AI companion activities

Activity 1. Check conventions

Ask an AI assistant:

Compute the complex inner product of $u=(1+i,2)$ and $v=(i,1-i)$. Use the convention $\langle u,v\rangle=v^*u$. Then explain what changes under the opposite convention.

Then verify the answer by hand.

Activity 2. Compare Schur and Jordan

Ask:

Why is Schur decomposition preferred over Jordan decomposition in numerical linear algebra?

A good answer should mention unitary matrices, conditioning, perturbations, and numerical stability.

Activity 3. Generate examples

Ask for one example of each:

Hermitian matrix;
unitary matrix;
normal but non-Hermitian matrix;
nonnormal matrix.

Then test each example in Python.

# Lab 11. Complex Inner Products, Unitary Matrices, Hermitian Matrices, and Schur Decomposition: Independent Study {.unnumbered} This lab accompanies **Chapter 11: Complex Inner Products, Unitary Geometry, Hermitian Matrices, and Schur Decomposition**. The goal is to connect complex geometry with computation: 1. Complex inner products require conjugation. 2. Unitary matrices preserve length and angle. 3. Hermitian matrices are the complex analogue of real symmetric matrices. 4. Schur decomposition triangularizes every complex square matrix using a unitary basis. 5. Normal matrices are exactly the matrices that can be unitarily diagonalized. This is an **independent-study lab**. Each main question includes a worked solution and a similar practice question. ## Python practice notebook You may use the Jupyter notebook version for longer Python practice: - [Download Lab 11 independent-study notebook](lab-11-complex-inner-products-schur-independent-study.ipynb) - [Open Lab 11 in Google Colab](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/lab-11-complex-inner-products-schur-independent-study.ipynb) ## Interactive lab ```{=html} <iframe src="lab-11-interactive.html" width="100%" height="4200" style="border:1px solid #ddd; border-radius:12px;"> </iframe> ``` ## Study guide and worked questions ### Question 1. Complex inner product Let $$ u=\begin{bmatrix}1+i\\2\end{bmatrix}, \qquad v=\begin{bmatrix}i\\1-i\end{bmatrix}. $$ Using the convention $\langle u,v\rangle=v^*u$, compute $\langle u,v\rangle$, $\langle v,u\rangle$, and $\|u\|$. #### Solution $$ \langle u,v\rangle =\overline{i}(1+i)+\overline{(1-i)}(2) =(-i)(1+i)+(1+i)2. $$ Since $$ (-i)(1+i)=1-i, \qquad 2(1+i)=2+2i, $$ we get $$ \langle u,v\rangle=3+i. $$ Therefore $$ \langle v,u\rangle=3-i, $$ and $$ \|u\|=\sqrt{|1+i|^2+|2|^2}=\sqrt6. $$ #### Similar practice Let $$ a=\begin{bmatrix}2-i\\1+i\end{bmatrix}, \qquad b=\begin{bmatrix}1+i\\-i\end{bmatrix}. $$ Compute $\langle a,b\rangle$, $\langle b,a\rangle$, and $\|a\|$. #### Answer $$ \langle a,b\rangle=\overline{(1+i)}(2-i)+\overline{(-i)}(1+i) =(1-i)(2-i)+i(1+i). $$ Now $$ (1-i)(2-i)=1-3i, \qquad i(1+i)=i-1. $$ Hence $$ \langle a,b\rangle=-2i, \qquad \langle b,a\rangle=2i. $$ Also $$ \|a\|=\sqrt{|2-i|^2+|1+i|^2}=\sqrt{5+2}=\sqrt7. $$ ### Question 2. Projection onto a complex line Let $$ u=\frac{1}{\sqrt2}\begin{bmatrix}1\\ i\end{bmatrix}, \qquad y=\begin{bmatrix}2\\1+i\end{bmatrix}. $$ Compute the projection of $y$ onto $\operatorname{span}(u)$. #### Solution Since $u$ is unit length, $$ \operatorname{proj}_{\operatorname{span}(u)}(y)=\langle y,u\rangle u. $$ Using $\langle y,u\rangle=u^*y$, $$ \langle y,u\rangle=\frac{1}{\sqrt2}\begin{bmatrix}1&-i\end{bmatrix} \begin{bmatrix}2\\1+i\end{bmatrix} =\frac{3-i}{\sqrt2}. $$ Thus $$ \operatorname{proj}(y)=\frac{3-i}{2}\begin{bmatrix}1\\ i\end{bmatrix}. $$ #### Similar practice Let $$ w=\frac{1}{\sqrt5}\begin{bmatrix}1+2i\\0\end{bmatrix}, \qquad z=\begin{bmatrix}3-i\\2\end{bmatrix}. $$ Compute $\operatorname{proj}_{\operatorname{span}(w)}(z)$. #### Answer Because $w$ is unit length, $$ \operatorname{proj}(z)=\langle z,w\rangle w. $$ Now $$ \langle z,w\rangle=w^*z =\frac{1}{\sqrt5}\overline{(1+2i)}(3-i) =\frac{1}{\sqrt5}(1-2i)(3-i) =\frac{1-7i}{\sqrt5}. $$ Therefore $$ \operatorname{proj}(z)=\frac{1-7i}{5} \begin{bmatrix}1+2i\\0\end{bmatrix}. $$ ### Question 3. Hermitian and unitary checks Let $$ A=\begin{bmatrix}2&1+i\\1-i&3\end{bmatrix}, \qquad U=\frac1{\sqrt2}\begin{bmatrix}1&1\\1&-1\end{bmatrix}. $$ Check whether $A$ is Hermitian and whether $U$ is unitary. #### Solution $$ A^*=\begin{bmatrix}2&1+i\\1-i&3\end{bmatrix}=A, $$ so $A$ is Hermitian. Also, $$ U^*U=I, $$ so $U$ is unitary. Since this matrix is real, $U^*=U^T$. #### Similar practice Check whether $$ B=\begin{bmatrix}1&i\\i&1\end{bmatrix} $$ is Hermitian. #### Answer $$ B^*=\begin{bmatrix}1&-i\\-i&1\end{bmatrix}\ne B. $$ So $B$ is not Hermitian. ### Question 4. Schur form and eigenvalues Suppose $$ A=UTU^*, \qquad T=\begin{bmatrix} 2&5&1\\0&3&4\\0&0&-1 \end{bmatrix}. $$ What are the eigenvalues of $A$? #### Solution Unitary similarity preserves eigenvalues, and the eigenvalues of an upper triangular matrix are the diagonal entries. Therefore the eigenvalues of $A$ are $$ 2,\quad 3,\quad -1. $$ #### Similar practice If $$ S=\begin{bmatrix} i&2\\0&1-i \end{bmatrix} $$ is a Schur form of a matrix $B$, what are the eigenvalues of $B$? #### Answer The eigenvalues are $$ i,\qquad 1-i. $$ ### Question 5. Normal but not Hermitian Give an example of a normal matrix that is not Hermitian. #### Solution Let $$ D=\begin{bmatrix}i&0\\0&1\end{bmatrix}. $$ Since $D$ is diagonal, it is normal: $$ D^*D=DD^*. $$ But $$ D^*=\begin{bmatrix}-i&0\\0&1\end{bmatrix}\ne D, $$ so $D$ is not Hermitian. #### Similar practice Show that $$ R=\begin{bmatrix}0&-1\\1&0\end{bmatrix} $$ is normal but not Hermitian. #### Answer Since $R$ is real orthogonal, $R^TR=RR^T=I$, so it is normal. But $$ R^T=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\ne R, $$ so it is not Hermitian. ## AI companion activities ### Activity 1. Check conventions Ask an AI assistant: > Compute the complex inner product of $u=(1+i,2)$ and $v=(i,1-i)$. Use the convention $\langle u,v\rangle=v^*u$. Then explain what changes under the opposite convention. Then verify the answer by hand. ### Activity 2. Compare Schur and Jordan Ask: > Why is Schur decomposition preferred over Jordan decomposition in numerical linear algebra? A good answer should mention unitary matrices, conditioning, perturbations, and numerical stability. ### Activity 3. Generate examples Ask for one example of each: 1. Hermitian matrix; 2. unitary matrix; 3. normal but non-Hermitian matrix; 4. nonnormal matrix. Then test each example in Python.