Eigendecomposition and "composition" in Julia

Question

I've been going through some tutorial on factorization in Julia. For the sake of practice, I am trying to take eigendecompositions from a matrix and recreate the original Matrix, using the formula:

A = VλV⁻¹

Where V is a matrix of eigenvectors, λ is a diagonal matrix of eigenvalues, and V⁻¹ is the inverted matrix V.

What confuses me is that the eigenvalues are returned as a vector, while the the guides I found states it should be returned as a diagonal matrix.

Code example:

using LinearAlgebra

# Create matrix
A = rand(3, 3)

# Eigendecomposition
AEig = eigen(A)

λ = AEig.values
 3-element Vector{Float64}:
 
V = AEig.vectors
 3×3 Matrix{Float64}:
  
Acomp = V*λ*inv(V)

A ≈ Acomp

Trying to multiply the vector and matrices returns an error:

DimensionMismatch("A has dimensions (3,1) but B has dimensions (3,3)")

This occurs because multiplying V with λ returns a 3-element vector, which is then attempted multiplied with V⁻¹, which is a 3×3 Matrix. My question is, is there a straightforward way to create a diagonal matrix from a vector? Alternatively, can "recomposition" of the original matrix be achieved another way?

Answer 1

You can use identity matrix from LinearAlgebra presented as I like this:

julia> λ                                                                                                                
3-element Vector{Float64}:                                                                                               
-0.4445656542213612                                                                                                      
0.5573883013610712                                                                                                      
1.310095519651262

julia> λ .* I(3)                                                                                                        
3×3 Matrix{Float64}:                                                                                                     
-0.444566  -0.0       -0.0                                                                                               
0.0        0.557388   0.0                                                                                               
0.0        0.0        1.3101 

The .* there means that every element of the vector is being multiplied by the corresponding row of the matrix.

[Edit:] I discovered another way to create a diagonal matrix, after posting the question using the Diagonal() function. While the above solution works, this creates somewhat simpler syntax:

julia> Diagonal(λ)
3×3 Diagonal{Float64, Vector{Float64}}:
 -0.444566   ⋅         ⋅ 
   ⋅        0.557388   ⋅ 
   ⋅         ⋅        1.3101