QR decomposition factorizes a matrix into an orthogonal matrix Q and an upper triangular matrix R. It is widely used for solving linear systems and eigenvalue problems.
⚠QR decomposition always exists for any matrix with linearly independent columns. It is numerically stable for solving linear systems.
What is QR Decomposition?
QR decomposition factorizes a matrix A into A = QR where Q is an orthogonal matrix (QᵀQ = I) and R is an upper triangular matrix. This decomposition is fundamental in numerical linear algebra.
Orthogonal Q
QᵀQ = I, columns are orthonormal basis vectors.
Upper Triangular R
R[i][j] = 0 for i > j, diagonal entries may be positive.
Numerical Stability
QR is numerically stable for solving linear systems.
Gram-Schmidt
Classical method using orthogonalization of columns.
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