What is use of diagonalization in control system?
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diagonal matrix is a matrix in which non-zero values appear only on its main diagonal. In other words, every entry not on the diagonal is 0. Diagonalization is the process of transforming a matrix into diagonal form.
A Diagonal Matrix
Not all matrices can be diagonalized. A diagonalizable matrix could be transformed into a diagonal form through a series of basic operations (multiplication, division, transposition and so on). However, this process can be long and is not easily described. Fortunately, diagonalization can be carried out through a more general algorithm that takes advantage of the matrix's characteristic polynomial.
Eigenvectors and Eigenvalues
Since eigenvectors and eigenvalues of a matrix are so important for understanding the how and why of a diagonal matrix, it would be worthwhile to quickly review them here. A matrix, say A, can be understood to represent a function or transformation that could be applied to a vector, say v. If the vector changes in magnitude, but does not otherwise transform, then it is an eigenvector of the matrix. Whatever value the vector is changed by is called the eigenvalue of the matrix. Usually the scalar number by which the vector is multiplied is called lambda. This relationship can be expressed through the following equation:

The diagonalized matrix is not in the same vector space as the original matrix. The eigenvectors of the matrix will be the basis of the new space. When a matrix has been diagonalized, the columns of each corresponds to an eigenvector of the matrix and each value (one per column) represent the eigenvalues of the matrix.
A Diagonal Matrix
Not all matrices can be diagonalized. A diagonalizable matrix could be transformed into a diagonal form through a series of basic operations (multiplication, division, transposition and so on). However, this process can be long and is not easily described. Fortunately, diagonalization can be carried out through a more general algorithm that takes advantage of the matrix's characteristic polynomial.
Eigenvectors and Eigenvalues
Since eigenvectors and eigenvalues of a matrix are so important for understanding the how and why of a diagonal matrix, it would be worthwhile to quickly review them here. A matrix, say A, can be understood to represent a function or transformation that could be applied to a vector, say v. If the vector changes in magnitude, but does not otherwise transform, then it is an eigenvector of the matrix. Whatever value the vector is changed by is called the eigenvalue of the matrix. Usually the scalar number by which the vector is multiplied is called lambda. This relationship can be expressed through the following equation:

The diagonalized matrix is not in the same vector space as the original matrix. The eigenvectors of the matrix will be the basis of the new space. When a matrix has been diagonalized, the columns of each corresponds to an eigenvector of the matrix and each value (one per column) represent the eigenvalues of the matrix.
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