After-lecture tweaking

Author: mfherbst

src/03_Matrix_eigenproblems.jl

@@ -284,8 +284,8 @@ Based on the quadratic form we can show
 > \begin{align}
 > 		&\nabla q_A(v) = \lambda \nabla g(v)
 > 		\\
-> 		&\begin{cases}
-> 			A^T \bar v  = \lambda \bar v & \qquad \text{(Wirtinger derivative wrt }x) \\ A v = \lambda v & \qquad \text{(Wirtinger derivative wrt } \bar x)
+> 		&(\ast) \begin{cases}
+> 			v^H A  = \lambda v^H & \qquad \text{(Wirtinger derivative wrt }x) \\ A v = \lambda v & \qquad \text{(Wirtinger derivative wrt } \bar x)
 > 		\end{cases}
 > \end{align}
 > ```
@@ -298,19 +298,15 @@ Our special interest are Hermitian matrices, particularly due to the importance
 As a reminder, Hermitian matrices satisfy $A^H = A$, i.e. $\langle u, Au \rangle = \langle Au, u \rangle$.
 For these, we have two key results :
 
-!!! note "Theorem 4"
+!!! note "Corollary 4"
 	Eigenvalues of Hermitian matrices are real.
 
 > *Proof.* 
-> If $(\lambda, u)$ is an eigenpair of Hermitian $A \in \mathbb C^{n \times n}$ with $\langle u, u \rangle = 1$, then
+> If $A = A^H$ we can rewrite the first condition $(\ast)$ in the above proof as
 > ```math
-> \begin{align}
-> 		\lambda &= \lambda \langle u, u \rangle
-> 		\\ &= \langle u, A u \rangle
-> 		\\ &= \langle A u, u \rangle
-> 		\\ &= \overline{\langle u, A u \rangle} = \bar \lambda &&& \square
-> \end{align}
+> v^H A^H = λ v^H \quad \Longleftrightarrow \quad A v = \bar \lambda v
 > ```
+> which in combination with the second condition of $(\ast)$ implies $\lambda = \bar \lambda$.
 """
 
 # ╔═╡ b7ad00f2-4112-458c-bf79-6c78a722fde2
@@ -343,7 +339,7 @@ md"""
 > V^H A V = \begin{pmatrix} \lambda_1 & C \\ 0 & B \end{pmatrix}
 > ```
 > where $B \in \mathbb C^{(n-1) \times (n-1)}, C \in \mathbb C^{1 \times (n-1)}$.
-> - As $W$ is an invariant subspace of $A$, so is $W^\perp$ by Lemma 5, and thus $C = 0$.
+> - Since $A$ is Hermitian and $V$ unitary, $V^H A V$ is Hermitian, which implies $B^H = B$ and $C = 0$.
 > - By induction hypothesis $B = Q D Q^T$ with $D$ real and diagonal and $Q$ unitary.
 >   The decomposition is thus obtained as 
 > ```math

src/04_Matrix_error_bounds.jl

@@ -99,8 +99,8 @@ md"""
 > ```math
 > 	M_{i j}= \begin{cases}0 & \text{if } i=j \\ \frac{A_{i j}}{A_{i i}-\lambda} & \text {otherwise }\end{cases}
 > ```
-> - Due to (2), we thus have $\sum_{j=1}^{n}\left|M_{i j}\right|<1$ $\forall i$, therefore $\|M\|_{\infty}< 1$, and $\|M\|_{2}<\|M\|_{\infty}<1$.
-> - Since $\|M\|_{2}$ bounds the modulus of the eigenvalues of $M$, we have that $I+M$ is non-singular, which implies $A-\lambda I$ is non-singular. Therefore $\lambda$ cannot be an eigenvalue of $A$.
+> - Due to (2), we thus have $\sum_{j=1}^{n}\left|M_{i j}\right|<1$ $\forall i$, therefore $\|M\|_{\infty}< 1$, and $\varrho(M) < \|M\|_{\infty}<1$.
+> - Since $\varrho(M)$ bounds the modulus of the eigenvalues of $M$, we have that $I+M$ is non-singular, which implies $A-\lambda I$ is non-singular. Therefore $\lambda$ cannot be an eigenvalue of $A$.
 > - This contradicts our initial statement.
 >   $\hspace{7cm} \square$
 
@@ -172,7 +172,7 @@ md"""
 >   $\hspace{12cm} \square$
 """
 
-# ╔═╡ fd5bcfb5-86c5-4097-a8b0-3f12ef8ed752
+# ╔═╡ bf7bc022-c5d2-40a1-ae48-731ee97d9a32
 md"""
 This is a simple way to get general error bound by establishing a **residual-error relationship**, i.e. a relation between the residual as a computable check for convergence and the error of our quantity of interest against the exact result. 
 We will note that there is no need to know the exact result !
@@ -190,8 +190,17 @@ We will note that there is no need to know the exact result !
 	Note that there is no reason for $q$ or $C$ to be identical in both cases.
 
 This suggests the following important point: error-residual relationships are not unique.
+"""
+
+# ╔═╡ a70cff6b-b3d6-4f49-aed0-4d01813d6a95
+md"""
+In fact in many practical examples such as [Herbst, Levitt, Cances 2020, *Figure 7*](https://michael-herbst.com/publications/2020.04.28_error_nonscf_kohn_sham.pdf) one sees that Bauer-Fike does not follow the convergence behaviour of the true error. 
+
 In fact for our case a better bound is the Kato-Temple bound, which we will derive next.
+"""
 
+# ╔═╡ 5e85fbbb-b9b6-49f2-b33f-539ddd7f4cc0
+md"""
 ## Kato-Temple bound
 """
 
@@ -463,7 +472,11 @@ Consider the near-diagonal matrix
 M = $(latexify_md(M))
 
 
-with some Sliders to tune the off-diagonal elements:
+with some Sliders to tune the off-diagonal elements, shown below. As approximate eigenvectors we assume the unit vectors, that is $\tilde{v}_i = e_i$. As a result the corresponding approximate eigenvalues are just
+```math
+	R_M(\tilde{v}_i) = \tilde{v}_i^T M \tilde{v}_i  = M_{ii},
+```
+i.e. the diagonal entries of $M$.
 """
 
 # ╔═╡ a7e14252-02ee-49c1-9f46-5699cf590923
@@ -1687,7 +1700,9 @@ version = "1.9.2+0"
 # ╟─9c189c48-497e-43c1-a50e-fc1ebe7f0714
 # ╟─371399ec-a571-4aed-b910-7307d90edad1
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-# ╟─fd5bcfb5-86c5-4097-a8b0-3f12ef8ed752
+# ╟─bf7bc022-c5d2-40a1-ae48-731ee97d9a32
+# ╟─a70cff6b-b3d6-4f49-aed0-4d01813d6a95
+# ╟─5e85fbbb-b9b6-49f2-b33f-539ddd7f4cc0
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