Question

Write your own code for the modified Gram-Schmidt algorithm, and apply your own code to have QR-decomposition of the matrix X, where X is given by:

1 1 1 1
1 2 4 8
1 3 9 ?

Answer 1

The Modified Gram-Schmidt algorithm is used for QR-decomposition of a matrix, creating an orthogonal matrix Q and an upper triangular matrix R. To perform the algorithm, you project and normalize each column vector of the matrix, and compute the upper triangular matrix R based on these projections.

Step-by-step explanation:

The Modified Gram-Schmidt algorithm is a numerical procedure to perform QR-decomposition of a matrix X. The following pseudocode represents the algorithm:

Applying this algorithm to the matrix X (which seems to have a typo but should represent a matrix with columns [1,1,1], [1,2,3], and [1,4,9]), you would obtain an orthogonal matrix Q and an upper triangular matrix R such that X = QR.

Keep in mind that in practice, this algorithm would be implemented using a programming language like Python, MATLAB, or R, and it would include additional steps to calculate the elements of R based on the dot products of ai with the obtained qj.