- #1
tennishaha
- 21
- 0
Homework Statement
v is a vector with norm(v)<1
what is the inverse of (I+vv') where I is a identity matrix
The inverse of (I+vv') is (I-vv'), where I is the identity matrix and v is a vector. This can be derived using the Sherman-Morrison formula: (I+uv')^-1 = I - (u/(1+v'u))v', where u and v are vectors.
The inverse of (I+vv') can be computed using the Sherman-Morrison formula or by using the matrix inversion method. This involves finding the determinant of the matrix and then using it to calculate the inverse.
In this problem, the norm of v being less than 1 ensures that the matrix (I+vv') is invertible. If the norm is greater than or equal to 1, the matrix will not have an inverse. This is because the determinant of (I+vv') is equal to 1-v'v, which becomes 0 if the norm of v is equal to 1.
Yes, the inverse of (I+vv') can be solved analytically using the Sherman-Morrison formula or the matrix inversion method. However, for larger matrices, it may be more efficient to use numerical methods.
The inverse of (I+vv') has various applications in mathematics, engineering, and computer science. It is used in solving systems of linear equations, computing determinants, and in machine learning algorithms such as linear regression and principal component analysis.