On The Frobenius Condition Number of Positive Definite Matrices

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Volume 2010, Article ID 897279,11pages doi:10.1155/2010/897279

Research Article

On The Frobenius Condition Number of

Positive Definite Matrices

Ramazan T ürkmen and Z übeyde Uluk ök

Department of Mathematics, Science Faculty, Selc¸uk University, 42003 Konya, Turkey

Correspondence should be addressed to Ramazan T ¨urkmen,[email protected]

Received 19 February 2010; Revised 4 May 2010; Accepted 15 June 2010

Academic Editor: S. S. Dragomir

Copyrightq2010 R. T ¨urkmen and Z. Uluk ¨ok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We present some lower bounds for the Frobenius condition number of a positive definite matrix depending on trace, determinant, and Frobenius norm of a positive definite matrix and compare these results with other results. Also, we give a relation for the cosine of the angle between two given real matrices.

1. Introduction and Preliminaries

The quantity

κA

⎧ ⎨ ⎩

AA−1 _if_{A is nonsingular,}

∞ ifAis singular 1.1

is called the condition number for matrix inversion with respect to the matrix norm·. Notice thatκA A−1_A_≥_A−1_A_I_≥_{1 for any matrix norm}_{see, e.g.,}₁_{, page 336}_{. The}

condition numberκA AA−1_{of a nonsingular matrix}_A_{plays an important role in the}

numerical solution of linear systems since it measures the sensitivity of the solution of linear systemsAxbto the perturbations onAandb. There are several methods that allow to find good approximations of the condition number of a general square matrix.

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whereA∗denotes the conjugate transpose ofA. A Hermitian matrixAis said to be positive semidefinite or nonnegative definite, written asA≥0, ifsee, e. g.,2, p.159

x∗_Ax_≥₀_, _∀_x_∈_Cn_, _1.2

Ais further called positive definite, symbolizedA >0, if the strict inequality in1.2holds for all nonzerox∈Cn. An equivalent condition forA∈Cn×nto be positive definite is thatA is Hermitian and all eigenvalues ofAare positive real numbers.

The trace of a square matrixAthe sum of its main diagonal entries, or, equivalently, the sum of its eigenvalues is denoted by trA. Let Abe any m×n matrix. The Frobenius

Euclideannorm of matrixAis

A_F

⎛ ⎝m

i1 n

j1

_a_ij 2

⎞ ⎠

1/2

. 1.3

It is also equal to the square root of the matrix trace ofAA∗, that is,

A_F √trAA∗. 1.4

The Frobenius condition number is defined byκFA AFA−1F. InRn×nthe Frobenius

inner product is defined by

A, B_F trATB 1.5

for which we have the associated norm that satisfiesA2_F A, A_F. The Frobenius inner product allows us to define the cosine of the angle between two given realn×nmatrices as

cosA, B A, BF

A_FB_F. 1.6

The cosine of the angle between two real n×nmatrices depends on the Frobenius inner product and the Frobenius norms of given matrices. Then, the inequalities in inner product spaces are expandable to matrices by using the inner product between two matrices.

Buzano in3obtained the following extension of the celebrated Schwarz inequality in a real or complex inner product spaceH; ·,·:

| a, x x, b| ≤ 1

2ab| a, b|x

2_, _1.7

for any a, b, x ∈ H. It is clear that for a b, the above inequality becomes the standard Schwarz inequality

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with equality if and only if there exists a scalarλ ∈K K RorCsuch thatx λa. Also Dragomir in4has stated the following inequality:

a, x x, b x2 −

a, b

2

≤ ab

2 , 1.9

wherea, b, x∈H, x /0. Furthermore, Dragomir4has given the following inequality, which is mentioned by Precupanu in5, has been showed independently of Buzano, by Richard in

6:

1

2 a, b − abx

2_≤_{a, x} _{x, b}_≤ 1

2 a, babx

2_. _1.10

As a consequence, in next section, we give some bounds for the Frobenius condition numbers and the cosine of the angle between two positive definite matrices by considering inequalities given for inner product space in this section.

2. Main Results

Theorem 2.1. LetAbe positive definite real matrix. Then

2 trA

detA1/n −n≤κFA, 2.1

whereκ_FAis the Frobenius condition number.

Proof. We can extend inequality1.9given in the previous section to matrices by using the Frobenius inner product as follows: LetA, B, X∈Rn×n. Then we write

A, XF X, BF

X2 F

− A, BF

2

≤ AFBF

2 , 2.2

where A, X_F trATX,and · _Fdenotes the Frobenius norm of matrix. Then we get

trATXtrXTB

X2 F

−tr

AT_B

2

≤ AFBF

2 . 2.3

In particular, in inequality2.3, if we takeBA−1_{, then we have}

trATXtrXTA−1

X2 F

−tr

AT_A−1

2

≤ AFA−1F

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Also, ifXandAare positive definite real matrices, then we get

trAXtrXA−1

X2 F

−n

2

≤ AFA−1F

2

κFA

2 , 2.5

whereκ_FAis the Frobenius condition number ofA.

Note that Dannan in7has showed the following inequality by using the well known arithmetic-geometric inequality, forn-square positive definite matricesAandB:

ndetAdetBm/n≤trAmBm, 2.6

wheremis a positive integer. If we takeAX,BA−1_{, and}_m_{1 in}_2.6_{, then we get}

ndetXdetA−11/n≤trXA−1. 2.7

That is,

n

detX detA

1/n

≤trXA−1. 2.8

In particular, if we takeXIin2.5and2.8, then we arrive at

trAtrA−1

n −

n

2

≤κFA,

n

1 detA

1/n

≤trA−1.

2.9

Also, from the well-known Cauchy-Schwarz inequality, sincen2_≤_tr_A_tr_A−1_{, one can obtain}

0< n≤2trAtrA

−1

n −n≤κFA. 2.10

Furthermore, from arithmetic-geometric means inequality, we know that

ndetA1/n≤trA. 2.11

Sincen≤trA/detA1/n, we write 0< n≤2trA/detA1/n−n. Thus by combining2.9

and2.11we arrive at

2 trA

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Lemma 2.2. LetAbe a positive definite matrix. Then

trA3/2_tr_A−1/2

trA −

n

2 ≥0. 2.13

Proof. Letλibe positive real numbers fori1,2, . . . , n. We will show that

_k

i1 λ3/2

i

_k

i1 λ−_i1/2

≥ k 2 _k i1 λi 2.14

for allk1,2, . . . , n. The proof is by induction onk. Ifk1,

λ3/2 1 ·λ

−1/2 1 λ1≥

1

2λ1. 2.15

Assume that inequality2.14holds for somek. that is,

_k

i1 λ3/2

i

_k

i1 λ−_i1/2

≥ k 2 _k i1 λi

. 2.16

Then

_k₁

i1 λ3/2

i

_k₁

i1 λ−_i1/2

_k

i1 λ3/2

i λ3k/12

_k

i1

λ−_i1/2λ−_k1₁/2

_k

i1 λ3/2

i

_k

i1 λ−_i1/2

k

i1

λ3/2

i λ−k11/2λi−1/2λ3k1/2

λk1

≥ k 2 k i1 λi k i1

λiλk1 λk1

≥ k

2

k

i1 λi1₂

k

i1

λiλk1 λk₂1

k1

2

_k₁

i1 λi

.

2.17

The first inequality follows from induction assumption and the inequality

a2_b2 ab ≥

ab

2 ≥

ab 2.18

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Theorem 2.3. LetAbe positive definite real matrix. Then

0≤2n trA

3/2

trAdetA1/2n −n≤κFA, 2.19

whereκ_FAis the Frobenius condition number.

Proof. LetX >0 andA >0. Then from inequality1.9we can write

A, XFX, A−1F

X2 F

−

A, A−1 F

2

≤ AFA−1F

2 2.20

where A, B_F trATBand · denotes the Frobenius norm. Then we get

trAXtrXA−1

X2 F

−n

2

≤ κFA

2 . 2.21

SetXA1/2_{. Then}

trA3/2_tr_A−1/2

trA −

n

2

≤ κFA

2 . 2.22

SincetrA3/2_tr_A−1/2_/_tr_A₋_n/₂_≥_{0 by}_{Lemma 2.2}_and_n_det_A−1/21/n_≤

trA−1/2,

trA3/2

trA n

detA−1/21/n− n 2 ≤

trA3/2_tr_A−1/2

trA −

n

2 ≤

κFA

2 . 2.23

Hence

2n trA

3/2

trAdetA1/2n −n≤κFA. 2.24

Let λi be positive real numbers fori 1,2, . . . , n. Now we will show that the left side of

inequality2.19is positive, that is,

2

n

i1 λ3/2

i ≥ _n i1 λi _n i1 λ1/2n

i

. 2.25

By the arithmetic-geometric mean inequality, we obtain the inequality

1 n _n i1 λi _n i1 λ1/2

i ≥ _n i1 λi _n i1 λ1/2n

i

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So, it is enough to show that

2

n

i1 λ3/2

i ≥ 1 n _n i1 λi _n i1 λ1/2

i

. 2.27

Equivalently, 2n n i1 λ3 i ≥ _n i1 λ2 i _n i1 λi

. 2.28

We will prove by induction. Ifk1, then

2λ3₁≥λ2₁·λ1λ3₁. 2.29

Assume that the inequality2.28holds for somek. Then

2k1

_k₁

i1 λ3 i 2k k i1 λ3

i 2 k

i1 λ3

i 2kλ3k12λ3k1

≥ _k i1 λ2 i _k i1 λi 2 _k i1 λ3

i λ3k1

2λ3_k₁

≥ _k i1 λ2 i _k i1 λi 2 k i1 λ2

iλk1λiλ2k1

2λ3_k₁

≥ _k i1 λ2 i _k i1 λi k i1 λ2

iλk1 k

i1

λiλ2_k₁λ3_k₁

_k₁ i1 λ2 i _k₁ i1 λi . 2.30

The first inequality follows from induction assumption and the second inequality follows from the inequality

a3_b3_≥_a2_b_ab2 _2.31

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Theorem 2.4. LetAandBbe positive definite real matrices. Then

cosA, IcosB, I≤ 1

2cosA, B 1. 2.32

In particular,

cosA, A−1≤cosA, IcosA−1, I≤ 1 2

cosA, A−11≤1. 2.33

Proof. We consider the right side of inequality1.10:

a, x x, b ≤ 1

2 a, babx

2_. _2.34

We can extend this inequality to matrices as follows:

A, X_F X, B_F ≤ 1

2 A, BF AFBFX

2

F 2.35

whereA, X, B∈Rn×n. Since A, X_FtrATX, it follows that

trATXtrXTB≤ 1 2

trATBA_FB_FX2_F, 2.36

LetXbe identity matrix andAandBpositive definite real matrices. According to inequality

2.36, it follows that

trAtrB≤ 1

2trABAFBFn, trAtrB

√

nA_F√nB_F ≤

1 2

trAB

AFBF 1

.

2.37

From the definition of the cosine of the angle between two given realn×nmatrices, we get

cosA, IcosB, I≤ 1

2cosA, B 1. 2.38

In particular, forBA−1_{we obtain that}

cosA, IcosA−1, I≤ 1 2

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Also, Chehab and Raydan in8have proved the following inequality for positive definite real matrixAby using the well-known Cauchy-Schwarz inequality:

cosA, A−1≤cosA, IcosA−1, I. 2.40

By combining inequalities2.39and2.40, we arrive at

cosA, A−1≤cosA, IcosA−1, I≤ 1 2

cosA, A−11 2.41

and since1/2cosA, A−1 ₁_n/₂_A

FA−1F 1/2andn ≤ κFA, we arrive at

1/2cosA, A−1 ₁_≤_{1. Therefore, proof is completed.}

Theorem 2.5. LetAbe a positive definite real matrix. Then

n√nA_F

trA ≤κFA. 2.42

Proof. According to the well-known Cauchy-Schwarz inequality, we write

_n

i1 λiA

2

≤

_n

i1 λ2

iA

n, 2.43

whereλiAare eigenvalues ofA. That is,

trA2≤ntrA2. 2.44

Also, from definition of the Frobenius norm, we get

trA≤√nA_F. 2.45

Then, we obtain that

cosA, I √trA

nA_F ≤1. 2.46

Likewise,

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When inequalities2.40and2.47are combined, they produce the following inequality:

cosA, A−1≤cosA, I,

n κFA ≤

trA

√

nA_F.

2.48

Therefore, finally we get

n√nA_F

trA ≤κFA. 2.49

Note that Tarazaga in9has given that ifAis symmetric matrix, a necessary condition to be positive semidefinite matrix is that trA≥ A_F.

Wolkowicz and Styan in10have established an inequality for the spectral condition numbers of symetric and positive definite matrices:

κ2A≥1 2s

m−s/p, 2.50

wherep√n−1,mtrA/n, ands A2_F/n−m21/2_.

Also, Chehab and Raydan in8have given the following practical lower bound for the Frobenius condition numberκ_FA:

κFA≥max

n,

√

n

cos2_{A, I},1

2s

m−s/p

. 2.51

Now let us compare the bound in2.49and the lower bound obtained by the authors in8

for the Frobenius condition number of positive definite matrixA. Since 0≤ A_F/trA≤1,A2_F/trA2≤ A_F/trA. Thus, we get

n√nA2 F

trA2 ≤

n√nA_F

trA ≤κFA. 2.52

All these bounds can be combined with the results which are previously obtained to produce practical bounds forκFA. In particular, combining the results given by Theorems

2.1,2.3, and2.5and other results, we present the following practical new bound:

κFA≥max

2 trA

detA1/n−n,2n

trA3/2

trAdetA1/2n −n,

n√nA_F

trA ,1 2s

m−s/p

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Example 2.6.

A

⎡ ⎢ ⎢ ⎣

4 1 0 2 1 5 1 2 0 1 6 3 2 2 3 8

⎤ ⎥ ⎥

⎦. 2.54

Here trA 23, A_F √179, detA 581, and have n 4. Then, we obtain that 2trA/detA1/n−n5.369444, 2ntrA3/2_/_tr_A_det_A1/2n₋_n₅_._741241,_n√_n_A

F/trA

4.653596, and 1 2s/m−s/p 2.810649. SinceκFA 6.882583, in this example, the best

lower bound is the second lower bound given byTheorem 2.3.

Acknowledgments

The authors thank very much the associate editors and reviewers for their insightful comments and kind suggestions that led to improving the presentation. This study was supported by the Coordinatorship of Selc¸uk University’s Scientific Research Projects.

References

1 R. A. Horn and C. R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985. 2 F. Zhang,Matrix Theory: Basic Results and Techniques, Springer, New York, NY, USA, 1999.

3 M. L. Buzano, “Generalizzazione della diseguaglianza di Cauchy-Schwarz,”Rendiconti del Seminario Matematico Universit`a e Politecnico di Torino, vol. 311971/73, pp. 405–409, 1974Italian.

4 S. S. Dragomir, “Refinements of Buzano’s and Kurepa’s inequalities in inner product spaces,”Facta Universitatis, no. 20, pp. 65–73, 2005.

5 T. Precupanu, “On a generalization of Cauchy-Buniakowski-Schwarz inequality,” Annals of the “ Alexandru Ioan Cuza” University of Ias¸i, vol. 22, no. 2, pp. 173–175, 1976.

6 U. Richard, “Sur des inegalites du type Wirtinger et leurs application aux equations diﬀerentielles ordinaries,” inProceedings of the Colloquium of Analysis, pp. 233–244, Rio de Janeiro, Brazil, August 1972.

7 F. M. Dannan, “Matrix and operator inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 2, no. 3, article 34, 2001.

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9 P. Tarazaga, “Eigenvalue estimates for symmetric matrices,”Linear Algebra and its Applications, vol. 135, pp. 171–179, 1990.