[Intro] Every matrix hides a secret shape Three transformations locked inside Rotate scale rotate back again The skeleton beneath the skin [Verse 1] You gave me rows and columns raw A cloud of numbers nothing more But something lives inside that grid A structure waiting to be freed Rotate the input find the frame Where all the action has a name Orthonormal clean and aligned The right singular vectors defined [Pre-Chorus] The singular values speak They rank the signal from the noise The loudest first the smallest last Geometry finds its voice [Chorus] Singular value decomposition Break it open see the truth U and sigma V transpose Three factors holding every proof Rotate then scale then rotate back The hidden frame comes into view Every matrix every map Has this decomposition too [Verse 2] The first singular vector points Along the direction of greatest spread The singular value gives the scale The second finds the next instead Orthogonal to what came before The axes fall in order clean Stack the values watch them drop The structure holds when they are gone [Pre-Chorus] The singular values speak They rank the signal from the noise The loudest first the smallest last Geometry finds its voice [Chorus] Singular value decomposition Break it open see the truth U and sigma V transpose Three factors holding every proof Rotate then scale then rotate back The hidden frame comes into view Every matrix every map Has this decomposition too [Bridge] Image compression rank one slice A single value times two vectors Add the next, the picture sharpens Add the rest and nothing differs The data had a skeleton We found it in the values Low rank approximation The noise was never signal [Verse 3] Least squares lives here pseudoinverse V sigma dagger U transpose Project the target onto column space The closest answer we can find The null space and the row space split The column space and its complement Every subspace has its place The four of them endure [Pre-Chorus] The singular values speak They rank the signal from the noise The loudest first the smallest last Geometry finds its voice [Final Chorus] Singular value decomposition Break it open see the truth U and sigma V transpose Three factors holding every proof Rotate then scale then rotate back The hidden frame comes into view Every matrix every map Has this decomposition too [Outro] Every matrix hides a frame Every frame reveals a shape Rotate scale rotate once more The skeleton was there before
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