The L2-spectral radius similarity is not transitive. Suppose that A1,…,Ar are m×m-matrices and B1,…,Br are real n×n-matrices. Then define ρ2(A1,…,Ar)=ρ(A1⊗A1+⋯+Ar⊗Ar)1/2. Then the generalized Cauchy-Schwarz inequality is satisfied:
ρ(A1⊗B1+⋯+Ar⊗Br)≤ρ2(A1,…,Ar)ρ2(B1,…,Br).
We therefore define the L2,d-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) as ∥(A1,…,Ar)≃(B1,…,Br)∥=ρ(A1⊗B1+⋯+Ar⊗Br)ρ2(A1,…,Ar)ρ2(B1,…,Br). One should think of the L2-spectral radius similarity as a generalization of the cosine similarity ⟨u,v⟩∥u∥⋅∥v∥ between vectors u,v. I have been using the L2-spectral radius similarity to develop AI systems that seem to be very interpretable. The L2-spectral radius similarity is not transitive.
∥(A1,…,Ar)≃(A1⊕B1,…,Ar⊕Br)∥=1 and
∥(B1,…,Br)≃(A1⊕B1,…,Ar⊕Br)∥=1, but ∥(A1,…,Ar)≃(B1,…,Br)∥ can take any value in the interval [0,1].
We should therefore think of the L2,d-spectral radius similarity as a sort of least upper bound of [0,1]-valued equivalence relations than a [0,1]-valued equivalence relation. We need to consider this as a least upper bound because matrices have multiple dimensions.
Notation: ρ(A)=limn→∞∥An∥1/n is the spectral radius. The spectral radius A is the largest magnitude of an eigenvalue of the matrix A. Here the norm does not matter because we are taking the limit.A⊕B is the direct sum of matrices while A⊗B denotes the Kronecker product of matrices.
The L2-spectral radius similarity is not transitive. Suppose that A1,…,Ar are m×m-matrices and B1,…,Br are real n×n-matrices. Then define ρ2(A1,…,Ar)=ρ(A1⊗A1+⋯+Ar⊗Ar)1/2. Then the generalized Cauchy-Schwarz inequality is satisfied:
ρ(A1⊗B1+⋯+Ar⊗Br)≤ρ2(A1,…,Ar)ρ2(B1,…,Br).
We therefore define the L2,d-spectral radius similarity between (A1,…,Ar) and (B1,…,Br) as ∥(A1,…,Ar)≃(B1,…,Br)∥=ρ(A1⊗B1+⋯+Ar⊗Br)ρ2(A1,…,Ar)ρ2(B1,…,Br). One should think of the L2-spectral radius similarity as a generalization of the cosine similarity ⟨u,v⟩∥u∥⋅∥v∥ between vectors u,v. I have been using the L2-spectral radius similarity to develop AI systems that seem to be very interpretable. The L2-spectral radius similarity is not transitive.
∥(A1,…,Ar)≃(A1⊕B1,…,Ar⊕Br)∥=1 and
∥(B1,…,Br)≃(A1⊕B1,…,Ar⊕Br)∥=1, but ∥(A1,…,Ar)≃(B1,…,Br)∥ can take any value in the interval [0,1].
We should therefore think of the L2,d-spectral radius similarity as a sort of least upper bound of [0,1]-valued equivalence relations than a [0,1]-valued equivalence relation. We need to consider this as a least upper bound because matrices have multiple dimensions.
Notation: ρ(A)=limn→∞∥An∥1/n is the spectral radius. The spectral radius A is the largest magnitude of an eigenvalue of the matrix A. Here the norm does not matter because we are taking the limit.A⊕B is the direct sum of matrices while A⊗B denotes the Kronecker product of matrices.