Original: The computation of the diagonal entries follows directly from Lemma 31.
Revised: The computation of the diagonal entries follows directly from Lemma 31.

Original: To see that all other entries of ∆ are zero, we need to show that no column of M has more than one non-zero entry.
Revised: To see that all other entries of ∆ are zero, we need to show that no column of M has more than one non-zero entry.

Original: So let us assume for contradiction that the kl -th column of M has non-zero entries in the ij -th row and in the ı -th row for ( i, j ) (cid:54) =
Revised: So let us assume for contradiction that the kl -th column of M has non-zero entries in the ij -th row and in the ı -th row for ( i, j ) (cid:54) =

Original: ( ı,  ) .
Revised: ( ı,  ) .

Original: If i, ı ∈ I , then Lemma 31 implies i
Revised: If i, ı ∈ I , then Lemma 31 implies i

Original: = i
Revised: = i

Original: k
Revised: k

Original: = ı
Revised: = ı

Original: and i j = l = i  , which contradicts ( i, j ) (cid:54) =
Revised: and i j = l = i  , which contradicts ( i, j ) (cid:54) =

Original: ( ı,  ) .
Revised: ( ı,  ) .

Original: If i, ı / ∈ I , we see from Lemma 31 that j = k =  and i =
Revised: If i, ı / ∈ I , we see from Lemma 31 that j = k =  and i =

Original: l = ı , which contradicts ( i, j ) (cid:54) =
Revised: l = ı , which contradicts ( i, j ) (cid:54) =

Original: ( ı,  ) .
Revised: ( ı,  ) .

Original: Finally, if i ∈ I and ı / ∈ I , then Lemma 31 yields that ı
Revised: Finally, if i ∈ I and ı / ∈ I , then Lemma 31 yields that ı

Original: = l = i j ∈
Revised: = l = i j ∈

Original: I ; a contradiction.
Revised: I ; a contradiction.

Original: Corollary 33.
Revised: Corollary 33.

Original: The characteristic polynomial of H is
Revised: The characteristic polynomial of H is

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Using Schur complements, we can compute the characteristic polynomial of H as follows:
Revised: Using Schur complements, we can compute the characteristic polynomial of H as follows:

Original: [Equation]
Revised: [Equation]

Original: By Corollary
Revised: By Corollary

Original: [Equation]
Revised: [Equation]

Original: the matrix ( t − 2)( tI rd y − D ) − ∆ is a diagonal matrix whose ij -th diagonal entry is ( t − 2)( t − D ij,ij ) − ∆ ij,ij .
Revised: the matrix ( t − 2)( tI rd y − D ) − ∆ is a diagonal matrix whose ij -th diagonal entry is ( t − 2)( t − D ij,ij ) − ∆ ij,ij .

Original: We write shortly δ ij :=
Revised: We write shortly δ ij :=

Original: ∆ ij,ij and use the identity D ij,ij = 2 σ 2 i j to further derive
Revised: ∆ ij,ij and use the identity D ij,ij = 2 σ 2 i j to further derive

Original: [Equation]
Revised: [Equation]

Original: The latter equality was derived by substituting speciﬁc values into the δ ij according to Corollary 32.
Revised: The latter equality was derived by substituting speciﬁc values into the δ ij according to Corollary 32.

Original: Rearranging the terms of this last expression of χ H ( t ) yields (19).
Revised: Rearranging the terms of this last expression of χ H ( t ) yields (18).

Original: Lemma 34.
Revised: Lemma 34.

Original: Let x, y > 0 .
Revised: Let x, y > 0 .

Original: The polynomial f ( t ) =
Revised: The polynomial f ( t ) =

Original: t 2 − 2 t ( x + 1) + 4( x − y ) has two real roots and at least one of them is positive.
Revised: t 2 − 2 t ( x + 1) + 4( x − y ) has two real roots and at least one of them is positive.

Original: Moreover, f ( t ) has a negative root if and only if x < y .
Revised: Moreover, f ( t ) has a negative root if and only if x < y .

Original: Proof.
Revised: Proof.

Original: The roots of f ( t ) are x + 1 ± (cid:112) ( x + 1) 2 − 4( x − y )
Revised: The roots of f ( t ) are x + 1 ± (cid:112) ( x + 1) 2 − 4( x − y )

Original: = x + 1 ± (cid:112) ( x − 1) 2 + 4 y .
Revised: = x + 1 ± (cid:112) ( x − 1) 2 + 4 y .

Original: So the discriminant is positive and f ( t ) has two real roots.
Revised: So the discriminant is positive and f ( t ) has two real roots.

Original: Clearly, one of these is positive.
Revised: Clearly, one of these is positive.

Original: The other one is negative if and only if x + 1 < (cid:112) ( x − 1) 2 + 4 y , which is equivalent to ( x + 1) 2 < ( x − 1) 2 + 4 y and thus to x < y .
Revised: The other one is negative if and only if x + 1 < (cid:112) ( x − 1) 2 + 4 y , which is equivalent to ( x + 1) 2 < ( x − 1) 2 + 4 y and thus to x < y .

Original: Proof of Theorem 30.
Revised: Proof of Theorem 30.

Original: It is left to count the number of negative roots of the univariate polynomial (19).
Revised: It is left to count the number of negative roots of the univariate polynomial (18).

Original: All the linear factors of (19) have non-negative roots.
Revised: All the linear factors of (18) have non-negative roots.

Original: The ij -th quadratic factor of (19), for i ∈
Revised: The ij -th quadratic factor of (18), for i ∈

Original: [ d y ] \ I and j ∈ I , has at most one negative root due to Lemma 34.
Revised: [ d y ] \ I and j ∈ I , has at most one negative root due to Lemma 34.

Original: Moreover, it has exactly one negative root
Revised: Moreover, it has exactly one negative root

Original: if and only if σ 2 j < σ 2 i , which is equivalent to j > i .
Revised: if and only if σ 2 j < σ 2 i , which is equivalent to j > i .

Original: Hence, the polynomial (19) has exactly # { ( j, i
Revised: Hence, the polynomial (18) has exactly # { ( j, i

Original: ) ∈
Revised: ) ∈

Original: I × [ d y ] \ I | j > i } many negative roots.
Revised: I × [ d y ] \ I | j > i } many negative roots.

Original: Theorem 12.
Revised: Theorem 12.

Original: If the singular values of Q 0 are pairwise distinct and positive, h Q 0 | M r has exactly (cid:0) mr (cid:1) critical points, namely the matrices Q
Revised: If the singular values of Q 0 are pairwise distinct and positive, h Q 0 | M r has exactly (cid:0) mr (cid:1) critical points, namely the matrices Q

Original: I = U Σ
Revised: I = U Σ

Original: I V T with #( I ) =
Revised: I V T with #( I ) =

Original: r .
Revised: r .

Original: Moreover, its unique local and global minimum is Q { 1 ,...,r } .
Revised: Moreover, its unique local and global minimum is Q { 1 ,...,r } .

Original: More precisely, the index of Q I as a critical point of h Q 0 | M r ( i.e. , the number of negative eigenvalues of the Hessian matrix for any local parameterization) is
Revised: More precisely, the index of Q I as a critical point of h Q 0 | M r ( i.e. , the number of negative eigenvalues of the Hessian matrix for any local parameterization) is

Original: [Equation]
Revised: [Equation]

Original: where I c = { 1 , . .
Revised: where I c = { 1 , . .

Original: , m } \ I .
Revised: , m } \ I .

Original: Proof.
Revised: Proof.

Original: This is an amalgamation of Corollary 29 and Theorem 30.
Revised: This is an amalgamation of Corollary 29 and Theorem 30.