$I$ is an $n$-by-$n$ identity matrix, $A$ is an $n$-by-$n$ precomputed symmetric positive-definite matrix.
Since $A$ is precomputed I may invert, factor, decompose, or do anything to $A$ before the algorithm starts.
However, there is a more effective way which takes into account the fact that we already have A and the input matrix was modified slightly.
The Sherman-Morrison formula serves as the basis of an algorithm: Here A is a square Nx N matrix whose inverse matrix we know, u and v are the columns of height N defining the matrix modification, (u These cases have a different computational complexity.
$k$ will converge (not monotonically) to the sought output.
Now, my question is if there is an efficient way to compute the inverse that does not involve computing the inverse of a full $n$-by-$n$ matrix? I'll just write it here if someone else has need for it: $(k I A)^=(k I PDP^)^=(P(D k I)P^)^=P(D k I)^P^$, where $A=PDP^$ is the eigenvalue decomposition.
Making a “rank-one update” to a matrix means adding to the matrix another matrix of rank one: This actually occurs quite a bit in statistics and optimisation – I’m interested in the topic because rank-one updates occur in certain applications of , but a perhaps more important case is in sequential regression problems.
Replace the i-th column (zero-based count) vector of Q with the i-th * column of the input matrix. * If it's zero, then the original matrix was not invertible. * * My CMatrix class allows for general m*n matrices, it has these members: * ldouble** x; // there's a typedef long double ldouble; in the header * UINT row, col; // row count, column count * * My CMat Tmp class is similar to CMatrix (it has the same members), * but it represents a temporal matrix used in internal calculations * * My CVector class allows for n-dimensional vectors, it has these members: * ldouble* x; * UINT dim; */ CMat Tmp CMatrix:: Det Inv(ldouble& det) const have a look at codeproject, uncancodenow, etc..
I am computing $(k I A)^$ in an iterative algorithm where $k$ changes in each iteration.
Using the Sherman-Morrison formula, update R (the inverse of Q). The Sherman-Morrison formula also updates the determinant of the matrix. If i = n, stop * * NOTES: * * This algorithm has the advantage of calculating the determinant of the original * matrix in the process.
Creating the Adjugate Matrix to Find the Inverse Matrix Using Linear Row Reduction to Find the Inverse Matrix Using a Calculator to Find the Inverse Matrix Community Q&A Inverse operations are commonly used in algebra to simplify what otherwise might be difficult.