Nov 5, 2015 · If all eigenvalues exist in the underlying field or ring and there are as many as the dimension is, there is clearly a basis of eigenvectors. This condition is sufficient but not necessary since diag(1,...,1) is diagonalized with just one (repeated) eigenvalue 1.

Apr 24, 2010 · The repeated eigenvalues problem is a mathematical concept that arises when attempting to find the eigenvalues and eigenvectors of a square matrix. It occurs when the matrix has one or more eigenvalues that have a multiplicity greater than one, meaning they appear more than once in the matrix's characteristic polynomial.

Dec 4, 2023 · Instead, such systems are put into Jordan canonical form, which includes Jordan blocks for the repeated eigenvalues. What is the Jordan canonical form and how is it used in solving systems with repeated eigenvalues? The Jordan canonical form is a block-diagonal matrix that simplifies the representation of a linear system with repeated eigenvalues.

Nov 9, 2004 · In summary, when dealing with repeated eigenvalues, you will need to find all the linearly independent eigenvectors associated with that eigenvalue to determine the basis. This basis may be different from the original basis, but it will still span the same subspace. I hope this helps clarify the process for finding the basis for repeated ...

Oct 1, 2006 · Diagonalization of a matrix with repeated eigenvalues is the process of finding a diagonal matrix that is similar to the original matrix, using eigenvectors. This is done by finding a basis of eigenvectors, which are vectors that do not change direction during a linear transformation, and using them to construct a new matrix that is in diagonal ...

May 24, 2011 · A and B are simultaneously diagonalizable if and only if there exist a specific matrix P such that both P^{-1}AP and P^{-1}BP are both diagonal) unless they are each diagonalizable separately. So you must assume, even though there are repeated eigenvalues, that there exist a basis consisting entirely of eigenvectors of, say, A.

May 6, 2016 · What are non-homogeneous systems with repeated eigenvalues? Non-homogeneous systems with repeated eigenvalues are systems of linear equations where the matrix has at least one eigenvalue with a multiplicity greater than one. This means that the eigenvalue appears multiple times in the characteristic equation, resulting in repeated roots.

Feb 16, 2012 · Solving for eigenvalues and eigenvectors allows us to understand the behavior and characteristics of a matrix. In the case of repeated eigenvalues, it helps us identify the presence of linearly dependent or redundant vectors in the matrix. What methods can be used to solve for eigenvalues and eigenvectors of a matrix with repeated eigenvalues?

May 7, 2016 · How do repeated eigenvalues affect the stability of a system? Repeated eigenvalues can indicate a lack of distinct eigenvalues in the system, which can lead to instability and unpredictable behavior. In these cases, additional analysis and techniques may be necessary to accurately model and control the system.

Sep 7, 2008 · A matrix will have repeated eigenvalues when the output space (Ax) is a lower dimension than the input space (x) (ie A takes any vector in R3 and puts in into a plane subspace of R3). So therefore we really only need 2 vectors to form a basis for the new space.