1. Complex inner product
Enter $u=(u_1,u_2)$ and $v=(v_1,v_2)$ using real and imaginary parts. We use $\langle u,v\rangle=v^*u$.
Vector $u$
+$i$
+$i$
Vector $v$
+$i$
+$i$
2. Why conjugation matters
For a complex vector $z$, the expression $z^Tz$ may fail to be positive, while $z^*z$ is always real and nonnegative.
3. Matrix checks
Enter a $2\times2$ complex matrix $A$. Use entries $a+bi$.
4. Unitary action on points
The plot shows a real slice of the action on the first two real coordinates. Unitary matrices preserve complex norm.
5. Schur form and eigenvalues
If $A=UTU^*$ and $T$ is upper triangular, then the eigenvalues of $A$ are the diagonal entries of $T$.
6. QR iteration experiment
This simplified QR iteration works best for real symmetric examples. It illustrates how repeated unitary similarity transformations move a matrix toward triangular/diagonal form.
7. Independent-study practice tasks with answers
Task A. Compute $\langle u,v\rangle$ for $u=(1+i,2)$ and $v=(i,1-i)$.
Answer: $\langle u,v\rangle=3+i$.
Answer: $\langle u,v\rangle=3+i$.
Task B. Decide whether $A=\begin{bmatrix}2&1+i\\1-i&3\end{bmatrix}$ is Hermitian.
Answer: Yes, because $A=A^*$.
Answer: Yes, because $A=A^*$.
Task C. Give a normal matrix that is not Hermitian.
Answer: $\begin{bmatrix}i&0\\0&1\end{bmatrix}$ is diagonal, hence normal, but not Hermitian.
Answer: $\begin{bmatrix}i&0\\0&1\end{bmatrix}$ is diagonal, hence normal, but not Hermitian.
Task D. If $T=\begin{bmatrix}2&5&1\\0&3&4\\0&0&-1\end{bmatrix}$ is a Schur form, find the eigenvalues.
Answer: $2,3,-1$.
Answer: $2,3,-1$.
Similar practice. If $S=\begin{bmatrix}i&2\\0&1-i\end{bmatrix}$ is a Schur form, find the eigenvalues.
Answer: $i$ and $1-i$.
Answer: $i$ and $1-i$.