Vitor Sousa

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Notebook

Eigendecomposition: What It Buys You

Derivation Math

A=QΛQ−1⇒At=QΛtQ−1A = Q \Lambda Q^{-1} \quad\Rightarrow\quad A^t = Q \Lambda^t Q^{-1}

For xt+1=Axtx_{t+1} = Ax_t, iterating t steps costs one exponentiation of a diagonal matrix instead of t matmuls.

Why it matters: Spectral radius ρ(A) = max|λᵢ| governs stability. |λ| < 1 contracts, |λ| > 1 explodes. This is why RNN gradients vanish or explode — it’s the same theorem.