General Principles

The usual form of the eigenvalue problem is written

$$\bss{A}\bi{x} = \alpha\bi{x}$$ (3.26)

where $\bss{A}$ is a square matrix $\bi{x}$ is an eigenvector and $\alpha$ is an eigenvalue. Sometimes the eigenvalue and eigenvector are called latent root and latent vector respectively. An $N\times N$ matrix usually has $N$ distinct eigenvalue/eigenvector pairs^3.2. The full solution of the eigenvalue problem can then be written in the form

$$\bss{A}\bss{U}_r = \bss{U}_r{\underline{\underline{\alpha}}}$$ (3.27)

$$\bss{U}_l\bss{A} = {\underline{\underline{\alpha}}}\bss{U}_l$$ (3.28)

where ${\underline{\underline{\alpha}}}$ is a diagonal matrix of eigenvalues and $\bss{U}_r$ ($\bss{U}_l$) are matrices whose columns (rows) are the corresponding eigenvectors. $\bss{U}_l$ and $\bss{U}_r$ are the left and right handed eigenvectors respectively, and $\bss{U}_l = \bss{U}_r^{-1}.$^3.3

• For Hermitian matrices $\bss{U}_l$ and $\bss{U}_r$ are unitary and are therefore Hermitian transposes of each other: $\bss{U}_l^\dagger = \bss{U}_r$.
• For Real Symmetric matrices $\bss{U}_l$ and $\bss{U}_r$ are also real. Real unitary matrices are sometimes called orthogonal.