Hermitian Matrix Calculator Online

A Hermitian matrix calculator is an online tool designed to determine if a matrix is Hermitian, meaning it is equal to its own conjugate transpose. This property is essential in many mathematical and physical theories, where symmetric matrices are a foundational element.

Formula of Hermitian Matrix Calculator

To determine if a matrix, A, is Hermitian, the calculator uses a simple yet powerful formula: A must be equal to A^H, its conjugate transpose. Here’s how you can understand this formula:

• A: Represents the matrix in question.
• A^H: Denotes the conjugate transpose of matrix A. For practical use, input matrix A into the calculator, which then computes A^H and checks if every element of A matches the corresponding element in A^H.

Table of General Terms and Relevant Calculations

Term	Definition	Example Calculation or Conversion
Matrix	A rectangular array of numbers organized in rows and columns.	For a 2×2 matrix: [ [a, b], [c, d] ]
Conjugate Transpose	The transpose of a matrix after taking the complex conjugate of each element.	If A = [ [1, 2i], [-2i, 3] ], then A^H = [ [1, -2i], [2i, 3] ]
Hermitian Matrix	A matrix that is equal to its conjugate transpose.	A matrix A is Hermitian if A = A^H.
Complex Conjugate	The change of the sign of the imaginary part of a complex number.	The complex conjugate of 3 + 4i is 3 – 4i.
Transpose	The matrix obtained by swapping rows with columns.	If A = [ [1, 2], [3, 4] ], then the transpose of A is [ [1, 3], [2, 4] ].

Example of Hermitian Matrix Calculator

Given Matrix:

Consider a 2×2 complex matrix A: A = [ [3, 2+i], [2-i, 1] ]

Step-by-Step Calculation:

1. Calculate the Conjugate Transpose (A^H):
   • The conjugate transpose of A, denoted as A^H, involves taking the transpose of A and then applying the complex conjugate to each element.
   • The transpose of A is [ [3, 2-i], [2+i, 1] ].
   • Applying the complex conjugate, A^H becomes [ [3, 2-i], [2+i, 1] ].
2. Check if A is Equal to A^H:
   • Compare each element of A with the corresponding element of A^H:
     • A[1,1] = 3 and A^H[1,1] = 3
     • A[1,2] = 2+i and A^H[1,2] = 2-i (not equal)
     • A[2,1] = 2-i and A^H[2,1] = 2+i (not equal)
     • A[2,2] = 1 and A^H[2,2] = 1
   • Since A[1,2] ≠ A^H[1,2] and A[2,1] ≠ A^H[2,1], matrix A is not Hermitian.

This example illustrates how a Hermitian matrix calculator can determine whether a given matrix exhibits the Hermitian property by comparing the matrix to its conjugate transpose.

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