QR decomposition viva ques practice

Viva questions on QR Decomposition

Q: What is QR decomposition?
A: QR decomposition is a method of decomposing a matrix into a product of two matrices: an orthogonal matrix Q and an upper triangular matrix R.

Q: What is an orthogonal matrix?
A: An orthogonal matrix is a square matrix whose columns are orthonormal, meaning that the dot product of any two distinct columns is zero and the dot product of any column with itself is one.

Q: What is an upper triangular matrix?
A: An upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero.

Q: What is the formula for QR decomposition?
A: Given a matrix A, QR decomposition can be expressed as A = QR, where Q is an orthogonal matrix and R is an upper triangular matrix. The elements of Q and R can be computed using the Gram-Schmidt process.

Q: What are some applications of QR decomposition?
A: QR decomposition can be used for solving systems of linear equations, computing least squares solutions to overdetermined systems, finding eigenvalues and eigenvectors, and for numerical optimization.

Q: Is QR decomposition unique?
A: No, QR decomposition is not unique. There are different variations of the QR decomposition that can produce different Q and R matrices, but they will always satisfy the equation A = QR.

Q: How is QR decomposition related to Gram-Schmidt orthogonalization?
A: QR decomposition can be computed using the Gram-Schmidt orthogonalization process, which involves orthonormalizing the columns of A to form the columns of Q and computing the elements of R from the resulting orthogonal basis.

Q: How is QR decomposition used in least squares regression?
A: In least squares regression, QR decomposition is used to find the solution to an overdetermined system of linear equations. By decomposing the matrix of coefficients into Q and R, it is possible to solve for the coefficients by back substitution, which is more efficient than directly computing the inverse of A.



Q: What is the QR decomposition?
A: The QR decomposition is a factorization of a matrix A into a product of two matrices, Q and R, where Q is an orthogonal matrix and R is an upper triangular matrix.

Q: How is the QR decomposition calculated?
A: The QR decomposition can be calculated using the Gram-Schmidt process or Householder reflections.

Q: What are the properties of the Q matrix in the QR decomposition?
A: The Q matrix in the QR decomposition is orthogonal, meaning that Q^T Q = I, where I is the identity matrix.

Q: What are the properties of the R matrix in the QR decomposition?
A: The R matrix in the QR decomposition is upper triangular, meaning that all the entries below the diagonal are zero.

Q: How can the QR decomposition be used to solve systems of linear equations?
A: The QR decomposition can be used to solve systems of linear equations by first computing the QR decomposition of the matrix A, and then solving the system using back substitution with the upper triangular matrix R.

Q: How can the QR decomposition be used to compute the least squares solution to an over-determined system of linear equations?
A: The QR decomposition can be used to compute the least squares solution to an over-determined system of linear equations by first computing the QR decomposition of the matrix A, and then solving the system using the back substitution method with the upper triangular matrix R.

Q: What is the condition number of a matrix, and how does it relate to the QR decomposition?
A: The condition number of a matrix is a measure of how sensitive the matrix is to changes in its input. The condition number of a matrix can be computed using the QR decomposition, and it can help determine the accuracy of the solution to a system of linear equations. A large condition number indicates that the matrix is ill-conditioned and small changes in the input can lead to large changes in the output.



Q: What is QR decomposition?
A: QR decomposition is a factorization of a matrix into a product of an orthogonal matrix and an upper triangular matrix.

Q: What is the significance of QR decomposition in linear algebra?
A: QR decomposition is a fundamental technique in linear algebra that is used in a wide range of applications, including solving systems of linear equations, computing eigenvalues and eigenvectors, and least squares approximation.

Q: How is QR decomposition computed?
A: QR decomposition can be computed using the Gram-Schmidt process, Householder reflections, or Givens rotations.

Q: What is the Gram-Schmidt process?
A: The Gram-Schmidt process is a method for orthonormalizing a set of linearly independent vectors in a vector space.

Q: What is the Householder reflection?
A: Householder reflection is a matrix transformation that reflects a vector about a plane or hyperplane defined by a unit vector.

Q: What is a Givens rotation?
A: A Givens rotation is a matrix transformation that rotates a vector in a plane defined by two unit vectors.

Q: What is the difference between QR decomposition and LU decomposition?
A: QR decomposition decomposes a matrix into an orthogonal matrix and an upper triangular matrix, while LU decomposition decomposes a matrix into a lower triangular matrix and an upper triangular matrix.

Q: What is the relationship between QR decomposition and eigenvalues?
A: QR decomposition can be used to compute eigenvalues and eigenvectors of a matrix, by iteratively applying QR decomposition to the matrix.

Q: What is the least squares approximation problem?
A: The least squares approximation problem is to find the best-fitting linear approximation of a set of data points that minimizes the sum of the squares of the residuals.

Q: How can QR decomposition be used to solve the least squares approximation problem?
A: QR decomposition can be used to solve the least squares approximation problem by expressing the system of equations as a matrix equation of the form Ax = b, where A is the matrix of coefficients, x is the vector of unknowns, and b is the vector of observed values. Then, QR decomposition is used to compute the matrix Q and R, and the solution x is obtained by solving the triangular system Rx = Q^T b.