Chapter 6: Geometric Transformation and Eigen Decomposition. In the chapter mind map this icon labels Spectral Decomp $A=V\Lambda V^{-1}$. The discussion below is excerpted and lightly edited from § Spectral Decomposition in Mathematics for AI and Machine Learning.

The spectral decomposition (also called spectral theorem for diagonalizable matrices) expresses a matrix as a weighted sum of rank-1 matrices formed from its eigenvectors:

where $\mathbf v_n$ are the right eigenvectors (columns of $V$), $\mathbf p_n$ are the left eigenvectors (rows of $V^{-1}$, or columns of $P = (V^{-1})^\top$), and $\lambda_n$ are the eigenvalues.

Each term $\lambda_n \mathbf v_n \mathbf p_n^\top$ is a rank-1 matrix that represents the contribution of the $n$-th eigenmode to the matrix $A$. The spectral decomposition reveals how $A$ acts on vectors: it projects onto each eigenvector direction, scales by the corresponding eigenvalue, and recombines the results.

Special case for symmetric matrices: When $A$ is symmetric, $\mathbf p_n = \mathbf v_n$ (left and right eigenvectors coincide), and the spectral decomposition simplifies to:

where the eigenvectors $\mathbf v_n$ form an orthonormal basis. This form is particularly important in Principal Component Analysis (PCA), where symmetric covariance matrices are decomposed into principal directions.