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Hermitian Matrix

 

Hermitian Matrix is a special type of  matrix, which is same as its conjugate transpose as expressed below.

 

 

In orther words, a Hermitian Matrix has following properties

  • The entries on the main diagonal are real.
  • The element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith columnEigenvalues of all Hermitian Matrix are all real
  • Eigenvalues of all Hermitian Matrix are all real

 

 

One example of Hermintian Matrix is as follows. In this example, if you conjugate the matrix A and then transpose it, the result is same as the original matrix (A). So you can say the matrix 'A' is a Hermitian matrix.

 

 

One of the important characteristics of Hermitian Matrix is that Eigenvalues of all Hermitian Matrix are all real as shown in the following example.

 

 

 

For more practical meaning of Hermitian matrix, refer to Visualizing Hermitian Matrix as An Ellipse with Dr. Geo (I strongly recommend you to read this)

 

 

Related Reading : Hermitian Conjugate, Conjugate Transpose