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Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?
Diagonalizable matrices have a simpler structure and are easier to work with in calculations. They also have special properties that make them useful in various applications, such as optimization problems and differential equations.
A square matrix is diagonalizable if it has n linearly independent eigenvectors, where n is the size of the matrix. This means that the matrix can be written as a diagonal matrix, with the eigenvalues of the matrix on the main diagonal.
No, not every matrix can be diagonalized. A matrix can only be diagonalized if it has n linearly independent eigenvectors, where n is the size of the matrix. This means that not all matrices have a simpler diagonal form.
Diagonalization is the process of finding a diagonal matrix that is similar to the original matrix. This diagonal matrix has the eigenvalues of the original matrix on the main diagonal. In other words, the eigenvalues are the entries on the main diagonal of the diagonalized matrix.
Diagonalization simplifies systems of linear equations by transforming them into a diagonal form. This makes it easier to solve for the unknown variables and find the solutions to the system. This is especially useful for large systems of equations, as diagonalization reduces the number of calculations needed.