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Hermitian matrix
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==Applications== Hermitian matrices are fundamental to [[quantum mechanics]] because they describe operators with necessarily real eigenvalues. An eigenvalue <math>a</math> of an operator <math>\hat{A}</math> on some quantum state <math>|\psi\rangle</math> is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues. In [[signal processing]], Hermitian matrices are utilized in tasks like [[Fourier analysis]] and signal representation.<ref>{{Cite web |last=Ribeiro |first=Alejandro |title=Signal and Information Processing |url=https://www.seas.upenn.edu/~ese2240/wiki/Lecture%20Notes/sip_PCA.pdf}}</ref> The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information. Hermitian matrices are extensively studied in [[linear algebra]] and [[numerical analysis]]. They have well-defined spectral properties, and many numerical algorithms, such as the [[Lanczos algorithm]], exploit these properties for efficient computations. Hermitian matrices also appear in techniques like [[singular value decomposition]] (SVD) and [[eigenvalue decomposition]]. In [[statistics]] and [[machine learning]], Hermitian matrices are used in [[Covariance matrix|covariance matrices]], where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.<ref>{{Cite web |title=MULTIVARIATE NORMAL DISTRIBUTIONS |url=https://dspace.mit.edu/bitstream/handle/1721.1/121170/6-436j-fall-2008/contents/lecture-notes/MIT6_436JF08_lec15.pdf}}</ref> Hermitian matrices are applied in the design and analysis of [[communications system]], especially in the field of [[multiple-input multiple-output]] (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties. In [[graph theory]], Hermitian matrices are used to study the [[Spectral graph theory|spectra of graphs]]. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.<ref>{{Cite web |last=Lau |first=Ivan |title=Hermitian Spectral Theory of Mixed Graphs |url=https://www.sfu.ca/~iplau/Edinburgh_CS_Project.pdf}}</ref> The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs.<ref>{{Cite journal |last1=Liu |first1=Jianxi |last2=Li |first2=Xueliang |date=February 2015 |title=Hermitian-adjacency matrices and Hermitian energies of mixed graphs |journal=Linear Algebra and Its Applications |language=en |volume=466 |pages=182β207 |doi=10.1016/j.laa.2014.10.028|doi-access=free }}</ref>
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