Spectral Representation

linear-algebra functional-analysis

Definition

Spectral Representation

A spectral representation is a way of describing a linear operator or matrix by decomposing it into parts associated with its eigenvalues and eigenvectors.

In finite dimensions, this often means writing the operator in a basis in which it becomes diagonal, or as a sum of projections onto eigenspaces.

Intuition

A spectral representation breaks an operator into the parts that act independently on each spectral component.

Examples

Example: Hermitian Operator

Spectral Representation

If $T:V\to V$ is self-adjoined, then $T$ has a spectral representation of form:

$$T=\sum _{i\ge 0}{\lambda }_i{P}_i$$

where ${\lambda }_i$ are the (real-valued) eigenvalues of of $T$ and each $P_i$ is an orthogonal projection.

Note that the convergence of this series is in the sense of the Hilbert-space norm, i.e., it holds that:

$$||Tx-\sum _{i=0}^k{\lambda }_i{P}_{i}x||\to 0\text{for }k\to \infty \forall x\in V$$ Link to original