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matematikuvol
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If ##\hat{A}\vec{X}=\lambda\vec{X}## then ##\hat{A}^{-1}\vec{X}=\frac{1}{\lambda}\vec{X}##
And what if ##\lambda=0##?
And what if ##\lambda=0##?
If X is an eigenvector of A with eigenvalue zero, then AX=0 and X≠0.
But if A is non-singular, then X=A−10=0.
For any matrix A, only one of the above can be true.
When ##\lambda=0##, it means that the matrix has a non-invertible or singular solution, as the determinant of the matrix is equal to zero. This has important implications in solving systems of linear equations and in understanding the behavior of the matrix.
When ##\lambda=0##, the eigenvectors of the matrix are in the null space or kernel of the matrix. This means that they are perpendicular to all other eigenvectors and are not affected by scalar multiplication. Essentially, they are the "zero vectors" of the matrix.
Yes, a matrix can have multiple zero eigenvalues. This occurs when the matrix has repeated rows or columns, resulting in repeated eigenvalues. In this case, there are multiple eigenvectors associated with the zero eigenvalue.
If a matrix has a zero eigenvalue, it cannot be diagonalized. This is because diagonalization requires all eigenvalues to be non-zero. In this case, the matrix is called defective and cannot be fully diagonalized.
A zero eigenvalue has many practical applications in fields such as physics, engineering, and computer science. It is used in solving systems of linear equations, analyzing the stability of systems, and in image processing. It also has important implications in quantum mechanics and the study of geometric transformations.