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Hermitian Matrices Calculator

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Example
Created on 2024-06-20Asked by Evelyn Wright (Solvelet student)
Determine whether the matrix A=(2i−i3) A = \begin{pmatrix} 2 & i \\ -i & 3 \end{pmatrix} is Hermitian.

Solution

To determine whether the matrix A=(2i −i3) A = \begin{pmatrix} 2 & i \ -i & 3 \end{pmatrix} is Hermitian: A matrix A A is Hermitian if A=A† A = A^\dagger , where A† A^\dagger is the conjugate transpose of A A . Calculate the conjugate transpose A† A^\dagger : A=(2i −i3) A = \begin{pmatrix} 2 & i \ -i & 3 \end{pmatrix} A†=(2−i i3) A^\dagger = \begin{pmatrix} 2 & -i \ i & 3 \end{pmatrix} Compare A A and A† A^\dagger : A=(2i −i3) A = \begin{pmatrix} 2 & i \ -i & 3 \end{pmatrix} A†=(2−i i3) A^\dagger = \begin{pmatrix} 2 & -i \ i & 3 \end{pmatrix} Since A=A† A = A^\dagger , the matrix A A is Hermitian. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Camila Harris on Solvelet
1. Find the eigenvalues and eigenvectors of the Hermitian matrix A=[3−2+i−2−i2]A = \begin{bmatrix} 3 & -2 + i \\ -2 - i & 2 \end{bmatrix},2. Show that the Hermitian matrix B=[4−3i3i2]B = \begin{bmatrix} 4 & -3i \\ 3i & 2 \end{bmatrix} has real eigenvalues and orthogonal eigenvectors.,
DefinitionA Hermitian matrix is a square matrix that is equal to the its own conjugate transpose. In other words, the transpose of a Hermitian square matrix is the same as its inverse. Hence, A is a Hermitian matrix if A=A∗, where A∗ is a conjugate transpose of A. For example, the matrix (2−i​i3​) is hermitian as A∗ = is (2i​−i3​), where the original matrix is equal to its conjugate transpose.
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