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A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.

The topic of bounds on eigenvalues of symmetric matrices has a long history (e.g. [

The present study is devoted to a new approach for bounding the largest eigenvalue of 3 × 3 correlation matrices. In Theorem 2.1 we derive some new optimal bounds by given determinant and trace of the squared correlation matrix. They are compared in Theorem 3.1 to the optimal bounds in [

Starting point is a real 3 × 3 matrix

where

Restricting the attention to correlation matrices

The set of correlation matrices is uniquely determined by the set of 3 upper diagonal elements

We ask for possibly optimal bounds for the largest EV (LEV) of a correlation matrix by given

Lemma 2.1. For all

In the following, we assume first that

Using that

In terms of

Inequality (I)

By Lemma 2.2 below the square root is always real. The lower bound is non-negative provided

Inequality (II)

The square root is real provided

is non-negative provided

Lemma 2.2. For all

Proof. Clearly, one has

Case 1:

One has

Case 2:

One has

How are the feasible inequalities (I) and (II) linked? Lemma 2.1 implies the inequalities

where the first and third inequalities are attained at the extreme points

Theorem 2.1. (Optimal bounds for the LEV of a 3 × 3 correlation matrix). The largest eigenvalue

Upper bound

Case (A):

The upper bound is attained at the extreme points

Lower bound

Case (B):

Sub-Case (B1):

Sub-Case (B2):

The lower bound is attained at the extreme points

Case (C1):

Sub-Case (C11):

Sub-Case (C12):

The lower bound

Case (C2):

Sub-Case (C21):

Sub-Case (C22):

The lower bound is not attained, but in the limit as

Remarks 2.1. If the bounds are attained, that is in the cases (A), (B) and (C1), they are the best bounds by given

information for 3 × 3 correlation matrices, a detailed comparison with the WS bounds is instructive and provided in Section 3. In contrast to this, for the same set of matrices with positive eigenvalues, the bounds in [

Proof of Theorem 2.1. It is clear by (2.6) and (2.8) that the upper bound in Case (A) must hold. Equality in (I) is attained when

which is defined when

tion is available when

The following result is about uniform bounds, which do not depend on the given information.

Corollary 2.1. (Uniform bounds for the LEV of a 3 × 3 correlation matrix). If

Proof. Clearly, the absolute maximum of value 3 in case (A) is attained when

Remark 2.2. The bounds also follow from the WS bounds in (3.1) of the next section. However, only the lower bound (B) tells us when it is attained.

For correlation matrices the WS bounds are optimal conditionally on the value of

which the one bounds are more stringent than the others. It is remarkable that for 3 × 3 correlation matrices the WS bounds yield actually contiguous bounds for all 3 EVs ( [

When refereeing to the bounds in (3.1), as function of

Theorem 3.1. The WS bounds compare with the bounds of Theorem 2.1 as follows:

Upper bound

With

Lower bound

Proof. A case by case analysis based on Theorem 2.1 and Equation (3.1) is required. In Case (A) one has

This quadratic polynomial in

With Lemma 3.1 below, and the proof of Theorem 3.1, this is only possible if

where

One has

The possible zeros of

Since

Lemma 3.1. For all

Proof. If

According to Theorem 3.1 the new bounds are more stringent than the WS bounds in the following cases: (Ac) and (B). Similar comparison statements can be made for other LEV bounds. For example, one can compare Theorem 2.1 with the MV bounds in [

compare the new lower bounds with the classical lower bound

in [

To conclude this study, it might be instructive to illustrate the results numerically. Since the LEV is the largest root of a cubic polynomial, a lot of formulas exist to calculate it. A most popular one is the exact trigonometric

Case | New Bound | WS Bound | ||
---|---|---|---|---|

(Aa) | (0.25, 0.25, 0) | 1.35355 | 2 | 1.40825 |

(Ab) | (−0.5, 0.5, −0.5) | 2 | 2.20711 | 2 |

(Ac) | (−0.5, 0.5, 0.5) | 1.5 | 1.5 | 2 |

(−0.5, 0.5, −1) | 2.36603 | 2.36603 | 2.41421 | |

(B) | (−0.5, 0.5, −1) | 2.36603 | 2.36603 | 1.70711 |

(−0.5, 0.5, 0.5) | 1.5 | 1.5 | 1.5 | |

(−0.5, 0.5, −0.5) | 2 | 1.5 | 1.5 | |

(C1) | (0.25, 0.25, −0.25) | 1.25 | 1.03229 | 1.25 |

(C2) | (0.5, 0.5, 0.49999) | 1.99999 | 1 | 1.5 |

Vieta formula, also known under Chebyshev cube root’s formula. Following [

Note that the first use of Vieta’s formulas for computing the eigenvalues of a 3 × 3 matrix is apparently due to [

Another quite recent and attractive evaluation of the LEV, which can be applied to correlation matrices of any dimension, is the limiting Bernoulli type ratio approximation formula in [