Jorma Kaarlo Merikoski

A best upper bound for the 2-norm condition number of a matrix

Abstract Let A be an n × n nonsingular real or complex matrix. The best possible upper bound for the ratio of the largest and smallest singular values of A, using tr A∗A , det A, and n only, is obtained. A comparison with an earlier bound is given, and the singular and nonsquare cases are included. If all the eigenvalues of A are real and positive, the best possible upper bound for the ratio of the largest and smallest eigenvalues of A, involving tr A, det A, and n only, is presented as well.

Characterizations and lower bounds for the spread of a normal matrix

Abstract The spread of an n×n matrix A with eigenvalues λ1,…,λn is defined by spr A = max j,k |λ j −λ k |. We prove that if A is normal, then spr A = max | x * Ax − y * Ay | | x , y ∈ C n ,∥ x ∥=∥ y ∥=1 = max | x * Ay + y * Ax | | x , y ∈ C n ,∥ x ∥=∥ y ∥=1, re x * y =0 = max | x * Ay + y * Ax | | x , y ∈ C n ,∥ x ∥=∥ y ∥=1, x * y =0 = max spr z A + z A * 2 | z∈ C ,|z|=1 = 2 max | x * A 2 x −( x * Ax ) 2 |+ x * A * Ax −| x * Ax | 2 1/2 | x ∈ C n ,∥ x ∥=1 . We also present several lower bounds for spr A , given by these characterizations.

Bounds for Eigenvalues Using the Trace and Determinant

Abstract Let A be a square matrix with real and positive eigenvalues λ 1 ⩾ … ⩾ λ n > 0, and let 1 ≤ k ≤ l ≤ n . Bounds for λ kl … λ l and λ k + … + λ l , involving k, l, n, tr A , and det A only, are presented.

Some Notes on de Oliveira's Determinantal Conjecture

Abstract Let A , B , U ∈ ℂ n x n , A = diag( a j ), B = diag( b j ), U unitary, D = det ⁡ ( A + U B U H ) , z σ = ∏ j = 1 n ( a j + b σ ( j ) ) , σ ∈ S n . De Oliveiras conjecture “ D is a convex combination of the z σ s” is shown to be true if all the z σ s are collinear. The question of representing | U ( m ) | 2 ( U ( m ) is the m th compound of U ) as a convex combination of the | P σ ( m ) |s is also studied. Here |⋅| and |⋅| 2 are understood elementwise, and P σ denotes the permutation matrix with rows corresponding to σ.

Bounds for ratios of eigenvalues using traces

Abstract Let A be an n × n matrix with real eigenvalues λ 1 ⩾ … ⩾ λ n , and let 1 ⩽ k l ⩽ n . Bounds involving tr A and tr A 2 are introduced for λ k / λ l , ( λ k − λ l )/( λ k + λ l ), and { kλ k + ( n − l + 1) λ l } 2 /{ kλ 2 k + ( n − l + 1) λ 2 l }. Also included are conditions for λ l >; 0 and for λ k + λ l > 0.

On a lower bound for the perron eigenvalue

A lower boundn−1Σi,kaik for the Perron eigenvalue of a symmetric non-negative irreducible matrixA=(aik) is studied and compared with certain other lower bounds.

On a conjecture of G. N. de Oliveira on determinants

Let A and B be complex diagonal n × A matrices with A = diag (a 1, …an )B = diag (b 1, …bn ). De Oliveiras conjecture where Co means the convex hull and S n is the symmetric group of degree n, is shown to be true in the case n = 3.

On operator norms of submatrices

Abstract Both of the following conditions are equivalent to the absoluteness of a norm ν in C n : (1) for all n × n diagonal matrices D =( d k ), the subordinate operator norm N ν ( D )=max k | d k |; (2) for all n × n matrices A , N ν ( A ) ⩽ N ν (| A |). These conditions are modified for partitioned matrices by replacing absolute values with norms of blocks. A generalization of absoluteness is thus obtained.

On the trace and the sum of elements of a matrix

It is demonstrated that in many situations the sum of elements and the trace of a matrix behave similarly.

Improving eigenvalue bounds using extra bounds

Abstract Bounds for various functions of the eigenvalues of a Hermitian matrix A , based on the traces of A and A 2 , are improved. A technique is presented whereby these bounds can be improved by combining them with other bounds. In particular, the diagonal of A , in conjunction with majorization, is used to improve the bounds. These bounds all require O ( n 2 ) multiplications.