Difference between revisions of "Hermitian matrix"
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==Properties of Hermitian matrices== | ==Properties of Hermitian matrices== | ||
| − | * The [[eigenvalue | + | * The [[eigenvalue]]s are all real. |
| − | * The [[eigenvector | + | * The [[eigenvector]]s corresponding to different eigenvalues are orthogonal. |
Because of these properties, Hermitian matrices have important applications in [[quantum mechanics]]. | Because of these properties, Hermitian matrices have important applications in [[quantum mechanics]]. | ||
[[Category:Linear algebra]] | [[Category:Linear algebra]] | ||
Revision as of 13:55, July 13, 2016
A Hermitian matrix is one that satisfies Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=M^\dagger} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M^\dagger} is the Hermitian conjugate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} (i.e., the matrix formed by transposing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and taking the complex conjugate of each element). As an example, the most general 2x2 Hermitian matrix has the form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} a & b \\ b^* & c \end{pmatrix} }
for arbitrary complex numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c} . In the case where all elements of the matrix are real, a Hermitian matrix becomes symmetric (as Hermitian conjugation then becomes equivalent to transposition).
Properties of Hermitian matrices
- The eigenvalues are all real.
- The eigenvectors corresponding to different eigenvalues are orthogonal.
Because of these properties, Hermitian matrices have important applications in quantum mechanics.