It has a significant application in Machine Learning

1. Sherman-Morrison Formula
Numerical Linear Algebra
Isaac Amornortey Yowetu
NIMS-GHANA
January 4, 2021

2. Sherman- Morrison Formula Proof
Outline
1 Sherman- Morrison Formula Proof
Isaac Amornortey Yowetu NIMS-GHANA
Sherman-Morrison Formula

3. Sherman- Morrison Formula Proof
Sherman- Morrison Formula Proof
Suppose that A is an invertible square matrix and u, v are
vectors. Given that 1 + vT
A−1
u = 0, then, Sherman-Morrison
Formula states that:
Formula
(A + uvT
)−1
= A−1
−
A−1
uvT
A−1
(1 + vT A−1u)
Isaac Amornortey Yowetu NIMS-GHANA
Sherman-Morrison Formula

4. Sherman- Morrison Formula Proof
Proof
Suppose
Y = A + uvT
X = (A + uvT
)−1
such that XY = YX = I
Case 1 where XY=I
XY = A−1
−
A−1
uvT
A−1
(1 + vT A−1u)
A + uvT
(1)
= A−1
A + A−1
uvT
−
A−1
uvT
A−1
A
1 + vT A−1u
−
A−1
uvT
A−1
uvT
1 + vT A−1u
(2)
Isaac Amornortey Yowetu NIMS-GHANA
Sherman-Morrison Formula

5. Sherman- Morrison Formula Proof
= A−1
A + A−1
uvT
−
(A−1
uvT
A−1
A + A−1
uvT
A−1
uvT
)
1 + vT A−1u
(3)
= A−1
A + A−1
uvT
−
(A−1
uvT
+ A−1
uvT
A−1
uvT
)
1 + vT A−1u
(4)
= I + A−1
uvT
−
A−1
u(vT
+ vT
A−1
uvT
)
1 + vT A−1u
(5)
= I + A−1
uvT
−
A−1
u(1 + vT
A−1
u)vT
1 + vT A−1u
(6)
= I + A−1
uvT
− A−1
uvT
(7)
= I (8)
Isaac Amornortey Yowetu NIMS-GHANA
Sherman-Morrison Formula

6. Sherman- Morrison Formula Proof
Proof
Case 2 where YX=I
YX = A + uvT
A−1
−
A−1
uvT
A−1
(1 + vT A−1u)
(9)
= AA−1
+ uvT
A−1
−
AA−1
uvT
A−1
1 + vT A−1u
−
uvT
A−1
uvT
A−1
1 + vT A−1u
(10)
= I + uvT
A−1
−
uvT
A−1
1 + vT A−1u
−
uvT
A−1
uvT
A−1
1 + vT A−1u
(11)
= I + uvT
A−1
−
u(vT
A−1
+ vT
A−1
uvT
A−1
)
1 + vT A−1u
(12)
Isaac Amornortey Yowetu NIMS-GHANA
Sherman-Morrison Formula

7. Sherman- Morrison Formula Proof
= I + uvT
A−1
−
u(1 + vT
A−1
u)vT
A−1
1 + vT A−1u
(13)
= I + uvT
A−1
− uvT
A−1
(14)
= I (15)
Since XY = YX = I, hence the proof of Sherman-Morrison
Formula
Isaac Amornortey Yowetu NIMS-GHANA
Sherman-Morrison Formula