11. Build projectors from an operator

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Work over the real numbers with

\[T=\begin{pmatrix}2&1\\0&3\end{pmatrix}. \]

Find two invariant directions

For eigenvalue \(2\), solving \((T-2I)v=0\) forces the second coordinate to zero. Choose \(v_2=(1,0)^{\mathsf T}\). For eigenvalue \(3\), solving \((T-3I)v=0\) gives equal coordinates. Choose \(v_3=(1,1)^{\mathsf T}\).

These vectors form a basis, and direct multiplication gives \(Tv_2=2v_2\) and \(Tv_3=3v_3\). The operator is diagonalizable.

Interpolate the selectors

The polynomial \(3-t\) equals one at \(2\) and zero at \(3\). The polynomial \(t-2\) does the opposite. Substitute the matrix into these polynomials:

\[P_2=3I-T=\begin{pmatrix}1&-1\\0&0\end{pmatrix},\qquad P_3=T-2I=\begin{pmatrix}0&1\\0&1\end{pmatrix}. \]

The identity

\[(T-2I)(T-3I)= \begin{pmatrix}0&1\\0&1\end{pmatrix} \begin{pmatrix}-1&1\\0&0\end{pmatrix}=0 \]

implies \(T^2-5T+6I=0\). An annihilating polynomial must vanish at both eigenvalues, so no degree-one polynomial suffices. Thus \((t-2)(t-3)\) is the minimal polynomial. Using the displayed identity,

\[P_2^2=9I-6T+T^2=3I-T=P_2,\qquad P_3^2=T^2-4T+4I=T-2I=P_3. \]

Also \(P_2P_3=P_3P_2=0\), \(P_2+P_3=I\) and \(T=2P_2+3P_3\).

Compute through the packets

For \(x=(2,3)^{\mathsf T}\), the components are \(P_2x=(-1,0)^{\mathsf T}\) and \(P_3x=(3,3)^{\mathsf T}\). Therefore

\[Tx=2(-1,0)^{\mathsf T}+3(3,3)^{\mathsf T}=(7,9)^{\mathsf T}. \]

These are the oblique projectors from Example 1. Their eigenspaces are not perpendicular in the ordinary inner product. Diagonalizability alone does not imply orthogonal spectral projectors.

Try it

Use the decomposition to calculate \(T^2x\).

Answer: \(T^2=4P_2+9P_3\), so \(T^2x=(23,27)^{\mathsf T}\). Direct multiplication gives

\[T^2=\begin{pmatrix}4&5\\0&9\end{pmatrix}. \]

Papers, proofs, Lean, and code

Terms in this example: Eigenvector · Eigenvalue · Eigenspace · Diagonalizable operator · Polynomial in an operator · Minimal polynomial · Spectral projector

The example uses a diagonalizable two-dimensional matrix. The works develop wider packet and spectral constructions with additional hypotheses.

Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates

Read · Written proofs · Lean coverage · Code and setup · Evidence

Related library guide and code examples. The guide covers related constructions and their software interfaces.

Python arithmetic check for this example: spectral_projectors(). This checks the displayed rational, modular or finite-set calculations exactly.

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