The inverse transformation of the matrix needs to be reversed, as follows,
The inverse of a transpose is equal to the transpose of the inverse.
After knowing the above basic knowledge, we perform matrix decomposition, such as LU decomposition and LDU decomposition, as follows,
Here, we first have a matrix A, and we perform basic row transformation on the matrix A to simplify it into a stepped matrix or a row-simplified matrix (the pivot is 1), that is, multiply the elementary matrix E by A to the left to obtain an upper triangular matrix U (U means Upper), and then we invert the elementary matrix and put it on the right side, which constitutes L (L mean Lower lower triangular matrix), which constitutes the LU decomposition, as follows,
So how many calculations do you need to do for a matrix row transformation (one multiplication + addition is one calculation), for an nxn matrix, the answer is about 1/3 of n to the third power.
permutations permutation matrix (allow row-to-row interchange), transpose transpose matrix. The permutations permutation matrix has a very magical example, that is, the inverse of the permutation matrix is equal to the transpose of the permutation matrix.