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In the next, R^n×n is represented for the set of n×n real matrices. For A∈R^n×n, let tr(A) denote the trace of A. Suppose A∈R^n×n be an random symmetric matrix, afterward we presume that the eigenvalues of A are organized so that λ₁(A) ≥ λ₂(A) ≥ ⋯ ≥ λ_n(A).

Lemma 1[19, p7-12]: If the components of x and y are nonnegative integers, then the following conditions are equivalent:

(i) x can be derived from y by a finite number of transfers.

(ii) the sum of the k largest components of x is less than or equal to the sum of the k largest components of y, k = 1, 2, ..., n, with equality when k = n.

(iii)

whenever each α_i > 0. Here Σ_π denotes summation over all permutations.

For x,y∈Rⁿ,

when x < y, x is said to be majorized by y. Then x is called weakly majorized by y and x is signaled by x < _w y. This notation and terminology was introduced by Hardy, Littlewood, and Polya [20]. The next lemmas are employed to demonstrate the principal results.

Lemma 2[21, p49]: If A, B ∈ R^n×n are symmetric matrices and 1 ≤ i₁ < ⋯ < i_k ≤ n, then for k = 1, 2, ⋯, n,

Especially, we have

with equality when k = n.

Lemma 3[21, p49]: If A, B ∈ R^n×n are symmetric matrices and 1 ≤ i₁ < ⋯ < i_k ≤ n, then for k = 1, 2, ⋯, n,

Especially, we have

with equality when k = n.

Lemma 4[21, p48]: If A, B ∈ R^n×n are symmetric positive semidefinite matrices and 1 ≤ i₁ < ⋯ < i_k ≤ n, then for k = 1, 2, ⋯, n,

Especially, we have

and

with equality when k = n.

Lemma 5[19, p160, B.7 eq(9)]: If x₁ ≥ ⋯ ≥ x_n, y₁ ≥ ⋯ ≥ y_n and x < _w y, then for any real array u₁ ≥ ⋯ ≥ u_n ≥ 0,

Lemma 6[12]: Let A ∈ R^n×n, and A = U^TΛU where U is orthogonal and Λ is diagonal with 0 ≤ λ_i(Λ) < 1. Then

Consider the discrete Lyapunov equation

For making up for the example that λ₁(AA^T) is not within the circle of radius 1, we should establish the next alternation. Applying the similarity conversion [6 eq.(14)]-[21 p56], we set

Then the modified Lyapunov equation is obtained

where λ1A~A~T<1.

Then the previous works for bound estimates is valid. Using equation (13) and the above lemmas, the next theorems can be attained.

Theorem 1: If the discrete Lyapunov equation (12) is applied and λ1A~A~T<1, then we attain the following inequality as

where E = U^-TU^-1 and U is the transformation matrix in equation (13).

<Proof> From Komaroff’s work [12], the solution to equation (14), using integer l, is

For notational convenience, set B~=A~A~T, then

Applying Lemma 2 to equation (16), in view of equation (17) and Lemma 4, we have

Considering λiB~<1, Lemma 6 can be employed in equation (18) to obtain

From P~=UTPU and Q~=UTQU, it is easy to know that

and

Then equation (19) can be rewritten as follows:

From

and

we can obviously recognize that

For notational convenience, set x_i = λ_i(E^-1P) and yi=λiE-1Q1-λiA~A~T-1. From equation (22) and (25), we know

Since λ₁(E) ≥ λ₂(E) ≥ ⋯ ≥ λ_n(E) ≥0, in view of Lemma 5, we have

Applying Lemma 4 to equation (27), we have

Note that 1-λiA~A~T=λn-i+1I-A~A~T. This completes the proof.

Corollary 1: For the discrete Lyapunov equation (12), if λ1A~A~T < 1, then we have

Furthermore, we have

<Remark> The bounds of equation (29) and (30) were presented in [3]. Therefore the result of [3] can be generalized by Theorem 1. And Theorem 1 also offers a improved bound estimate of the solution.

Theorem 2: For the discrete Lyapunov equation (11), we have

<Proof> Using the arithmetic-mean geometric-mean inequality [19, p125-126] on equation (15) leads directly to the above bound. If λ_i(P) > 0 for i = 1, ..., n, then

equation (32) leads equation (15) to equation (31) as follows:

In this section, by utilizing the majorization transformation, some lower bound estimates of the solution can be obtained for the discrete Lyapunov equation without the presumption of λ₁(AA^T) < 1.

Theorem 3: Let the positive definite matrix P be the solution to equation (12). If λ₁(AA^T) < 1, then we have

<Proof> For P~=A~TP~A~+Q~, we have

By Lemma 3, we obtain

Applying Lemma 4 to equation (36), we have

From equation (37), we know

Then

Considering λnA~A~T<1, from equation (20) and (21), equation (39) can be rewritten as follows:

In view of Lemma 4, we have

Applying equation (41) to equation (40), we obtain

when k = n, we have

Note that λ1E-1=λn-1E. This completes the proof.

Consider the discrete Lyapunov equation (12) with system matrix A in [6 eq.(42)]

A=0.75630.85640.3536-0.7856,Q=1001.

Then the eigenvalues of AA^T are given by

In order to overcome the difficulty that λ₁(AA^T) is unstable, we introduce the similarity transformation matrix U. For this example, we choose U=1.93590.20450.2312-1.1352. Afterward we can have the Jordan-converted matrix and its eigenvalues as

Now, we eliminate the assumption of λ1A~A~T<1. Using Theorem 1, we have

Using Theorem 2, we obtain the following determinant bound

From Theorem 3, we have

Using Theorem 1 of [3], we have

Using Theorem 2 of [3], we have

By comparison, we know equation (46) is better than equation (49) and (50).

Consider the discrete Lyapunov equation (12) with

A=0.25390.581000.8958,Q=1001.

Then the eigenvalues of AA^T are given by

In order to overcome the difficulty that λ₁(AA^T) is unstable, we introduce the similarity transformation matrix U. For this example, we choose U=0.92320.07870.2141-0.2140. Afterward we can have the Jordan-converted matrix and its eigenvalues as

Now, we eliminate the assumption of λ1A~A~T<1. Using Theorem 1, we have

By Theorem 2, we obtain the following determinant bound

From Theorem 3, we have

Using Theorem 1 of [3], we have

Using Theorem 2 of [3], we obtain

By comparison, we know equation (53) is better than equation (56) and (57).