QR Algorithm

Definition

The QR algorithm is a numerical method used to compute the eigenvalues and eigenvectors of a matrix by decomposing it into an orthogonal matrix Q and an upper triangular matrix R. This algorithm is particularly important in the study of eigenvalues and diagonalization, as it provides a systematic approach to approximate these values iteratively, leading to convergence for many types of matrices.

5 Must Know Facts For Your Next Test

1. The QR algorithm iteratively applies QR factorization to the matrix and updates it, ultimately leading to a matrix that converges to an upper triangular form where the eigenvalues can be easily read off.
2. The convergence properties of the QR algorithm are dependent on the initial choice of the matrix, with symmetric or Hermitian matrices often leading to faster convergence.
3. When dealing with large matrices, the QR algorithm is preferred due to its numerical stability compared to other methods like the power iteration method.
4. The QR algorithm can be modified using shifts (known as the QR algorithm with shifts) to improve convergence speed, particularly for matrices that are not well-conditioned.
5. In addition to finding eigenvalues, the QR algorithm can also be used to compute eigenvectors through back-substitution once the eigenvalues are known.

Review Questions

• How does the QR algorithm utilize QR factorization to find eigenvalues?
  • The QR algorithm uses QR factorization by decomposing a given square matrix A into an orthogonal matrix Q and an upper triangular matrix R. By replacing A with the product RQ in subsequent iterations, the process gradually transforms A into an upper triangular form. The eigenvalues can then be extracted from this upper triangular matrix once the iterations converge, as the diagonal elements represent the eigenvalues.
• Compare the efficiency of the QR algorithm with other numerical methods for computing eigenvalues.
  • The QR algorithm is generally more efficient than methods like the power iteration method, especially for large matrices. This efficiency arises from its ability to maintain numerical stability while converging to eigenvalues through iterative updates. In contrast, methods like power iteration may converge slowly or even fail to find all eigenvalues unless combined with other techniques. The choice of method often depends on the properties of the matrix being analyzed.
• Evaluate how modifying the QR algorithm with shifts affects its convergence properties and performance.
  • Modifying the QR algorithm with shifts significantly enhances its convergence properties by allowing it to avoid stagnation during iterations. By incorporating shifts, which adjust the spectrum of the matrix being analyzed, the algorithm can accelerate convergence towards the actual eigenvalues. This adjustment helps especially in cases where matrices are poorly conditioned or have close eigenvalues. Overall, using shifts allows for more efficient computations and leads to faster results in practical applications.

Related terms

Eigenvalue: A scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation represented by a matrix.

Diagonalization: The process of converting a matrix into a diagonal form, where all off-diagonal elements are zero, which simplifies many matrix computations.

Orthogonal Matrix: A square matrix whose rows and columns are orthogonal unit vectors, which means its transpose is equal to its inverse.