[Verse 1] In Hilbert spaces vast and wide Lives an operator, self-adjoint pride Symmetric cousin of matrices known But infinite dimensions it calls home Real spectrum waiting to unfold Like eigenvalues, but stories untold [Chorus] Spectral theorem breaks it down Projections dancing all around Integral over lambda's range Diagonalize what once seemed strange E of lambda, measure true Multiplication's breaking through [Verse 2] Every observable in quantum land Self-adjoint operators, measurement's hand Eigenvalues are outcomes we see Eigenvectors, states that could be From finite sums to integrals flowing Continuous families, ever growing [Chorus] Spectral theorem breaks it down Projections dancing all around Integral over lambda's range Diagonalize what once seemed strange E of lambda, measure true Multiplication's breaking through [Bridge] Transform the abstract, make it clear Multiplication operator's here In L-two space where functions dwell Scalar lambda tells the tale Yes-no measurements in array Build reality in spectral way [Verse 3] From vibrations to signals processed clean Natural frequencies in the machine Quantum fields and matrices random Spectral geometry, understanding's anthem One truth echoes through the math Operators yield to spectrum's path [Chorus] Spectral theorem breaks it down Projections dancing all around Integral over lambda's range Diagonalize what once seemed strange E of lambda, measure true Multiplication's breaking through [Outro] Self-adjoint means spectrum's real Projection measures help us feel The infinite made tangible Through spectral decomposition's call
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