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Hilbert spaces and the spectral theorem

Inner products, orthonormal bases, Fourier series, and eigenvalues of operators.

An inner product gives angles and projections; an orthonormal basis expands any vector — Fourier series are exactly this in L². Self-adjoint operators have real spectrum and an eigen-decomposition, the infinite-dimensional version of diagonalising a symmetric matrix. Picture it: the symmetric matrix [[2,1],[1,2]] stretching along two perpendicular eigenvectors. Think it: quantum observables are self-adjoint operators; measurement outcomes are their spectrum.

Kugwira ntchito chitsanzo: `eigenvalues of [[2,1],[1,2]]`

Linear-Algebra · Eigenvalues

Eigenvalues of [[2,1],[1,2]]

\left[\begin{matrix}2 & 1\\1 & 2\end{matrix}\right]

Gawo ndi Gawo

\det(A - \lambda I) = 0
Eigenvalues are the roots of the characteristic polynomial.
\det\left[\begin{matrix}2 - \lambda & 1\\1 & 2 - \lambda\end{matrix}\right] = 0
Subtract λ from the diagonal.
\lambda^{2} - 4 \lambda + 3 = 0
Expand the determinant.
\left(\lambda - 3\right) \left(\lambda - 1\right) = 0
Factor.
\lambda = 3, \lambda = 1
Eigenvalues (with multiplicity).
\lambda = 1:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}-1\\1\end{matrix}\right]
Solve (A − 1I)v = 0 for a basis eigenvector.
\lambda = 3:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}1\\1\end{matrix}\right]
Solve (A − 3I)v = 0 for a basis eigenvector.

Kusonyeza yankho

\lambda = 3,\; \lambda = 1

Symbols used here

\det A,\ |A|

determinant
Scaling factor of area/volume under A; zero means singular.

\lambda

lambda (eigenvalue)
The factor by which an eigenvector is stretched: Av = λv.

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

matrix
A rectangular array of numbers; a linear map.

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow

not, and, or, implies, iff
Logical connectives.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|

dot product, norm
Σ u_i v_i; the length of v, √(v·v).

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\|f\|_p,\ \langle f, g \rangle

p-norm, inner product
Length of a function; the generalised dot product.

How to: Hilbert spaces and the spectral theorem

Eigenvalues are the roots of the characteristic polynomial.
Subtract λ from the diagonal.
Expand the determinant.
Factor.
Eigenvalues (with multiplicity).
Solve (A − 1I)v = 0 for a basis eigenvector.
Solve (A − 3I)v = 0 for a basis eigenvector.

Questions people ask

What is a Hilbert space?

A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.

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Zambiri pa Functional Analysis

Normed and Banach spaces

Kugwira ntchito chitsanzo: eigenvalues of [[2,1],[1,2]]