42. Diagonal Matrix

A square matrix whose only non-zero entries lie on the main diagonal:

Often written as $\text{diag}\left({𝑑}_1,{𝑑}_2,\dots ,{𝑑}_{𝑛}\right)$.

42.1. Why diagonal matrices are useful

Operations become trivial — they act componentwise:

• Multiplication by vector: $𝐷\overrightarrow{𝑥}=\left({𝑑}_1{𝑥}_1,{𝑑}_2{𝑥}_2,\dots ,{𝑑}_{𝑛}{𝑥}_{𝑛}\right)$
• Powers: ${𝐷}^{𝑘}=\text{diag}\left({𝑑}_1^{𝑘},\dots ,{𝑑}_{𝑛}^{𝑘}\right)$
• Inverse (when each ${𝑑}_{𝑖}\ne 0$): ${𝐷}^{-1}=\text{diag}\left(\frac{1}{{𝑑}_1},\dots ,\frac{1}{{𝑑}_{𝑛}}\right)$
• Determinant: $det\left(𝐷\right)={𝑑}_1{𝑑}_2\dots {𝑑}_{𝑛}$
• Trace: $\text{tr}\left(𝐷\right)={𝑑}_1+{𝑑}_2+\dots +{𝑑}_{𝑛}$
• Eigenvalues: ${𝑑}_1,{𝑑}_2,\dots ,{𝑑}_{𝑛}$ themselves
• Eigenvectors: standard basis vectors ${𝑒}_1,\dots ,{𝑒}_{𝑛}$

42.2. Special cases

• Identity matrix: all ${𝑑}_{𝑖}=1$
• Zero matrix (square): all ${𝑑}_{𝑖}=0$
• Scalar matrix: all ${𝑑}_{𝑖}=𝑐$ for the same scalar $𝑐$ (i.e. $𝑐𝐼$)

42.3. Where they show up

• Diagonalization: turning a general matrix into a diagonal one via change of basis
• SVD: $𝐴=𝑈\mathrm{\Sigma }{𝑉}^{𝑇}$ where $\mathrm{\Sigma }$ is diagonal
• Eigen-decomposition: $𝐴=𝑃𝐷{𝑃}^{-1}$ where $𝐷$ is diagonal of eigenvalues