Chapter 4

[[lecture-main-topic-data]]

In this chapter, Courant-Fisher is the main central theorem.

• a special case is Rayleigh-Ritz, which we proved by using that the field of values for a normal matrix is the convex hull of the spectrum
• this also gave us Weyl’s Theorem, a continuity result about adding a perturbing matrix and how this changes the eigenvalues of the original matrix.

• We also used this to get interlacing theorem 1 which gives us bounds on the eigenvalues of sums of hermitian matrices

• and interlacing 2, which gives us bounds on the eigenvalues of principal submatrices.
  • this gave us that the diagonal elements of a hermitian matrix majorize its eigenvalues
  • and also the sum of the first least eigenvalues is the minimum of the trace of orthonormal multiplications
  • this also gave us Hadamard’s Inequality, which is the same as the above diagonal elements of a hermitian matrix majorize its eigenvalues

Concepts

Concepts

TABLE status, type, filter(file.inlinks, (i) => contains(i.type, "lecture_note")) as Lectures
FROM #concepts
WHERE contains(parent, this.file.link)
OR contains(parent, "TOPIC")
SORT file.name asc
 
FLATTEN choice(type="theorem", "🧮","") + choice(type="instruction", "📓","") + choice(!type, "💡", "") as type

Mentions

Mentions

TABLE
FROM [[]]
 
FLATTEN choice(contains(artist, this.file.link), 1, "") + choice(contains(author, this.file.link), 1, "") + choice(contains(director, this.file.link), 1, "") + choice(contains(source, this.file.link), 1, "") + choice(contains(parent, this.file.link), 1, "") as direct_source
 
WHERE !direct_source

return function View() {
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	const modified = file.value("modified")
	
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}