Eigenvectors and associated eigenvalues are important in vibration analysis, differential  equations in all physical and biological sciences, statistics (factor analysis and principal components analysis in psychological and social sciences) and many other applied mathematical areas.

Eigendecomposition or sometimes called spectral decomposition is the factorization of a matrix into a canonical form.

Spectral Decomposition Theorem. Let A ∈ Mn with eigenvalues λ₁,λ₂,…,λn. Then A is normal (i.e., real symmetric, Hermitian) if and only if A is unitarily diagonalizable, that is, there exists a unitary matrix U such that

                                U∗AU = diag(λ₁,λ₂,…,λn).

There are a lot of applications of diagonalizable matrices in linear algebra and matrix theory. For examples: 

- Principal Axis Theorem. "An n×n real matrix A is orthogonally diagonalizable if and only if A is symmetric". This is a special case of spectral decomposition.

 - If A is Hermitian, then all the eigenvalues of A are real.

Thank you very much Prof. Wiwat Wanicharpichat, Dr.

Majid Forghani-elahabad and Dr. James Leigh your answers were very useful to me.

Dear Chathranee:

The most important application Eigen-value Eigen-vector decomposition is the decorrelation  of your data or matrix. In turn, it can be used in the reduction of the dimensionality of your data. For example, the Eigen-value Eigen-vector decomposition or PCA is used to determine or select the most dominant band/bands in multi-spectral or hyper-spectral remote sensing.

Regards,

Gamal

Spectral value decomposition for thin bed seismic imaging stratigraphic traps hydrocarbon exploration

It is used in car design especially car stereo system and also in decoupling three phase system.

Eigendecomposition is particularly useful for analysing the structure of the data matrix in terms of the eigenvalues and eigenvectors.

The eigendecomposition process is integral to many machine learning algorithms such as PCA and DMD.