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Can anyone prove that the eigenvectors of symmetric matrices are orthogonal?
An eigenvector of a symmetric matrix is a vector that, when multiplied by the matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector remains unchanged, but its magnitude is scaled by a factor known as the eigenvalue.
Eigenvectors and eigenvalues are closely related in a symmetric matrix. The eigenvalues represent the scaling factor for each eigenvector, and the set of all eigenvectors and their corresponding eigenvalues form a basis for the matrix.
Eigenvectors of symmetric matrices have numerous applications in mathematics, physics, and engineering. They are used in solving systems of linear equations, diagonalizing matrices, and in various machine learning algorithms.
The most common method for finding the eigenvectors of a symmetric matrix is by using the eigenvalue decomposition (EVD) method. This involves finding the eigenvalues and corresponding eigenvectors through a series of calculations.
Yes, a symmetric matrix can have complex eigenvectors. However, for a symmetric matrix, the eigenvalues are always real numbers. This means that if a complex eigenvector exists, its complex conjugate must also be an eigenvector.