16 Eigen Decomposition
Today we will discuss Eigen Decomposition. we know that A V is equal to lambda V, where lambda is eigen value, and v is eigen vector.
A is d x d matrix. There d eigen values λ1, λ2… λd and there are d eigen value v1, v2… vd.
When A matrix has certain structures where eigen values will start decreasing. we can write the above equations as
A [v1, v2…. vd] = [λ1v1, λ2v2… λdvd]
AV = VΛ
Where Λ is Diagonal matrix
A = VΛ(V inverse)
When, A is real and symmetric, Λ is orthogonal and V inverse becomes V transpose
A = VΛ(V transpose)
How useful is above equation?
To find the A inverse
A = VΛ(Vtranspose)
A inverse = (VΛ(Vtranspose)) inverse
A inverse = V (Λ inverse) (V transpose)
Where (Λ inverse) is diagonal matrix ( 1/λ1, 1/λ2… 1/λd)
A = sum(λi Vi Vt) (for i 1 to d), if λ are decreasing order
A = λ1[ ] + λ2[ ] +… +λd[ ]
A prime = λ1[ ] + λ2[ ]
This is helpful for lower rank approximation of the input matrix.
Reference: https://www.mathsisfun.com/algebra/eigenvalue.html
