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HomeMatlabWhat Is the Matrix Inverse? – Nick Higham

# What Is the Matrix Inverse? – Nick Higham

The inverse is a crucial theoretical software, however it’s hardly ever essential to compute it explicitly. If we want to resolve a linear system of equations $Ax = b$ then computing $A^{-1}$ after which forming $x = A^{-1}b$ is each slower and fewer correct in floating-point arithmetic than utilizing LU factorization (Gaussian elimination) to unravel the system instantly. Certainly, for $n = 1$ one wouldn’t resolve $3x = 1$ by computing $3^{-1} times 1$.

For sparse matrices, computing the inverse could not even be sensible, because the inverse is often dense.

If one must compute the inverse, how ought to one do it? We’ll think about the price of totally different strategies, measured by the variety of elementary arithmetic operations (addition, subtraction, division, multiplication) required. Utilizing (1), the price is that of computing one determinant of order $n$ and $n^2$ determinants of order $n-1$. Since computing a $ktimes k$ determinant prices at the least $O(k^3)$ operations by normal strategies, this strategy prices at the least $O(n^5)$ operations, which is prohibitively costly until $n$ could be very small. As an alternative one can compute an LU factorization with pivoting after which resolve the techniques $Ax_i = e_i$ for the columns $x_i$ of $A^{-1}$, at a complete value of $2n^3 + O(n^2)$ operations.

Equation (2) doesn’t give a very good technique for computing $A^{-1}$, as a result of computing the coefficients $c_i$ and evaluating a matrix polynomial are each costly.

It’s attainable to use quick matrix multiplication strategies, which compute the product of two $ntimes n$ matrices in $O(n^alpha)$ operations for some $alpha < 3$. Through the use of a block LU factorization recursively, one can cut back matrix inversion to matrix multiplication. If we use Strassen’s quick matrix multiplication technique, which has $alpha = log_2 7 approx 2.807$, then we will compute $A^{-1}$ in $O(n^{2.807})$ operations.

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