Normal matrices

A complex square matrix A is a normal matrix if

$A^{\dagger }A=AA^{\dagger },$

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A^{\dagger }} is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [A,A^{\dagger }]=0} .

Normal matrices are precisely those to which the spectral theorem applies: a matrix $A$ is normal if and only if it can be represented by a diagonal matrix $\Lambda$ and a unitary matrix $U$ by the formula

$A=U\Lambda U^{\dagger },$

where

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Lambda =\mathrm {diag} (\lambda _{1},\lambda _{2},\dots )}

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle U^{\dagger }U=UU^{\dagger }=I.}

The entries $\lambda _{i}$ of the diagonal matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda} are the eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} , and the columns of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} are the eigenvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . The matching eigenvalues in $\Lambda$ must be ordered as the eigenvectors are ordered as columns of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} .

References

• Normal matrix entry in Wikipedia