QR decomposition is a numerical method used to decompose a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). This decomposition can be used to solve a variety of problems, including finding eigenvalues of a matrix. By decomposing a matrix into its QR form, the eigenvalues can be easily calculated from the diagonal entries of the upper triangular matrix.
To use QR decomposition to find eigenvalues, first decompose the matrix into its QR form using a numerical algorithm such as Gram-Schmidt or Householder. Then, use the diagonal entries of the upper triangular matrix to determine the eigenvalues. These diagonal entries will be the same as the eigenvalues of the original matrix.
No, QR decomposition can only be performed on square matrices. This is because the decomposition requires the matrix to be multiplied by its transpose, which can only be done for square matrices.
One limitation of using QR decomposition to find eigenvalues is that it may not always be the most efficient method. In some cases, other numerical methods such as power iteration or Jacobi method may be faster and more accurate. Additionally, QR decomposition may not work for matrices with complex eigenvalues.
The accuracy of the eigenvalues found through QR decomposition depends on the accuracy of the decomposition itself. If the decomposition is performed accurately, the eigenvalues will also be accurate. However, if there are errors in the decomposition, the eigenvalues will also be affected.