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Difference between revisions of "Normal matrix"

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A square [[Matrix|matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067540/n0675401.png" /> that commutes with its adjoint (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067540/n0675402.png" />).
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A square complex [[matrix]] $A$ that commutes with its [[adjoint matrix]] $A^*$: that is, $AA^*=A^*A$.
  
 
====Comments====
 
====Comments====
See also [[Normal operator|Normal operator]].
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See also [[Normal operator]].
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The [[eigenvector]]s of a normal matrix form an [[orthonormal system]].  A matrix $A$ is normal if and only if it is unitarily [[Similar matrices|similar]] to a [[diagonal matrix]]: $\Delta = U^{-1} A U$ with $u$ a [[unitary matrix]].
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====References====
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* Lloyd N. Trefethen, David Bau III, ''Numerical Linear Algebra'' SIAM (1997) {{ISBN|0898713617}}
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[[Category:Special matrices]]

Latest revision as of 05:42, 22 April 2023

A square complex matrix $A$ that commutes with its adjoint matrix $A^*$: that is, $AA^*=A^*A$.

Comments

See also Normal operator.

The eigenvectors of a normal matrix form an orthonormal system. A matrix $A$ is normal if and only if it is unitarily similar to a diagonal matrix: $\Delta = U^{-1} A U$ with $u$ a unitary matrix.

References

  • Lloyd N. Trefethen, David Bau III, Numerical Linear Algebra SIAM (1997) ISBN 0898713617
How to Cite This Entry:
Normal matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_matrix&oldid=11980