A note on similarity to contraction for stable $2 \times 2$ companion matrices
We consider companion matrices of size $2 \times 2$ with general complex spectra satisfying a root condition with respect to the closed complex unit circle or the closed left complex half plane. For both cases, smooth and naturally conditioned basis transformations are constructed such that the resu...
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| author | Auzinger, W. Аузінгер, В. |
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| description | We consider companion matrices of size $2 \times 2$ with general complex spectra satisfying a root condition with respect to the closed complex unit circle or the closed left complex half plane. For both cases, smooth and naturally conditioned basis transformations are constructed such that the resulting, transformed matrix is contractive or dissipative, respectively, with
respect to the $\ell_2$-norm. |
| first_indexed | 2026-03-24T02:13:48Z |
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К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
W. Auzinger (Inst. Anal. und Sci. Comput., Techn. Univ. Wien, Austria)
A NOTE ON SIMILARITY
TO CONTRACTION FOR STABLE \bftwo \times \bftwo COMPANION MATRICES
ПРО ПОДIБНIСТЬ ВIДНОСНО СТИСКУ ДЛЯ СТАБIЛЬНИХ
\bftwo \times \bftwo СУПУТНIХ МАТРИЦЬ
We consider companion matrices of size 2\times 2 with general complex spectra satisfying a root condition with respect to the
closed complex unit circle or the closed left complex half plane. For both cases, smooth and naturally conditioned basis
transformations are constructed such that the resulting, transformed matrix is contractive or dissipative, respectively, with
respect to the \ell 2-norm.
Розглядаються супутнi матрицi розмiром 2 \times 2, загальнi комплекснi спектри яких задовольняють кореневу умову
вiдносно замкненого комплексного одиничного кола або замкненої лiвої комплексної напiвплощини. В обох ви-
падках будуються гладкi базиснi природно обумовленi перетворення такi, що результуюча перетворена матриця є
стискаючою або дисипативною вiдносно \ell 2-норми, вiдповiдно.
1. Introduction. For discrete or continuous evolution processes of the form
\bfity \nu +1 = A\bfity \nu , \nu \geq 0, or \bfity \prime (t) = A\bfity (t), t \geq 0, with A \in \BbbC n\times n,
the asymptotic behavior for \nu \rightarrow \infty or t \rightarrow \infty is determined by the location of the spectrum of A,
while the initial, transient behavior is governed by \| A\| or \mu (A), respectively, where \mu (A) denotes
the logarithmic matrix norm. It is well known that, for nonnormal A significant transient growth can
occur even if the system has a stable spectrum. The questions of describing the transient behavior, or
of bounding the evolution operator \| A\nu \| or \| etA\| uniformly in \nu or t, respectively, has been studied
in many papers on linear stability theory.
One of the classical results on this topic is the Kreiss Matrix Theorem (see, e.g., [4, 6] and
references therein), which involves several equivalent conditions on the matrix A which in turn are
equivalent to uniform boundedness of families of evolution operators in the \ell 2-norm \| \cdot \| 2. All these
equivalent conditions are not constructive and usually difficult to verify. Therefore it is a relevant
question in what cases auch bounds can be derived in a more or less explicit manner, depending on
the spectrum and making use of certain additional information about the matrix A.
A closer inspection of the literature indeed reveals that results of this type are naturally restricted
to cases for which additional structural properties are known (or assumed). A very special case is the
family of 2\times 2 companion matrices C \in \BbbC 2\times 2 describing the evolution of the BDF2 approximation to
scalar ODEs y\prime (t) = \lambda y(t). This method is A-stable, and a uniform, well-conditioned transformation
is known such that the transformed matrix is contractive for arbitrary Re(\lambda ) \leq 0 and h > 0. This is
a direct consequence of the G-stability of the scheme, which is equivalent to A-stability; see [4] for
details. Higher order A(\alpha )-stable BDF schemes have been considered in [2]: Here, the distribution of
the spectrum is analyzed, and combination with the resolvent condition in the Kreiss matrix theorem
leads to growth bounds uniformly valid with respect to the stability domain of the scheme. Some
further results of related type can, e.g., be found in [3] and [5].
Here we consider families of 2 \times 2 companion matrices with arbitrary complex stable spectra
with respect to the unit circle or the left half plane, respectively. We construct a natural basis
c\bigcirc W. AUZINGER, 2016
400 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
A NOTE ON SIMILARITY TO CONTRACTION FOR STABLE 2\times 2 COMPANION MATRICES 401
transformation such that the transformed matrix behaves contractive or dissipative, respectively, with
respect to \| \cdot \| 2. This basis transformation should feature a natural conditioning behavior and depend
smoothly on the spectrum.
At first sight this problem seems to be rather simple. However, from our results it can be seen
that already the case n = 2 is technically rather intricate. We provide a solution for both cases,
namely contractivity and dissipativity. Our results include a quantitative measure for the “distance
to instability”, defined in terms of the location of the spectrum. These results and their proofs are
specified in Sections 3 and 4. Examples are also given.
The construction used in the proofs of Propositions 3.1 and 4.1 cannot be directly extended to
dimension n \geq 3 due to the significantly more complicated algebra involved. (Some numerical
solutions for numerical data are, however, given in [1].) For more general classes of matrices,
explicit, quantitative results of this type seem to be very hard to obtain.
In the sequel, for a square matrix S, S > 0 means that S is positive definite (analogously for
<, \geq , \leq ).
2. Problem setting. Consider
C =
\Biggl(
0 1
- c0 - c1
\Biggr)
=
\Biggl(
0 1
- \zeta 1 \zeta 2 \zeta 1 + \zeta 2
\Biggr)
\in \BbbC 2\times 2 (2.1a)
with characteristic polynomial
\pi (\zeta ) = \zeta 2 + c1 \zeta + c0 = (\zeta - \zeta 1)(\zeta - \zeta 2) . (2.1b)
We study the problem of finding a basis transformation, preferably well-conditioned, converting a
given companion matrix C with a [weakly] stable spectrum into an \ell 2-contractive, or \ell 2-dissipative
matrix, respectively. The underlying assumption is that the spectrum of C satisfies a [weak] stability
condition w.r.t. the closed complex unit circle or the closed complex left half plane.
To this end one may first think of proceeding from the Jordan form of C,
C = XJX - 1. (2.2)
For \zeta 1 \not = \zeta 2 the matrix C is diagonalizable with eigensystem represented by a Vandermonde matrix V,
X = V =
\Biggl(
1 1
\zeta 1 \zeta 2
\Biggr)
, J =
\Biggl(
\zeta 1 0
0 \zeta 2
\Biggr)
.
For \zeta 1 = \zeta 2, however, the matrix C is not diagonalizable. We have (2.2) with X lower diagonal,
X = L =
\Biggl(
1 0
\zeta 1 1
\Biggr)
, J =
\Biggl(
\zeta 1 1
0 \zeta 1
\Biggr)
.
This discontinuous behavior of the Jordan form in the limit \zeta 1 \rightarrow \zeta 2 makes it inappropriate for our
purpose. As an alternative, one may proceed from a similarity transformation of C which valid for
arbitrary \zeta 1, \zeta 2 and continuous in the limit \zeta 1 \rightarrow \zeta 2, namely (2.2) with
X = L =
\Biggl(
1 0
\zeta 1 1
\Biggr)
, J =
\Biggl(
\zeta 1 1
0 \zeta 2
\Biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
402 W. AUZINGER
In order transform C to a form which is \ell 2-contractive or \ell 2-dissipative, respectively, for any
stable spectrum \{ \zeta 1, \zeta 2\} , it turns out that yet another, “symmetric” modification of such a similarity
transformation is more useful. Furthermore, we need to introduce a scaling parameter \delta which later
will be chosen in an appropriate way. In particular, we consider a transformation of C of the form
C = LT L - 1, (2.3a)
with
L =
\biggl(
1 0
\mu \delta
\biggr)
, T =
\left( \mu \delta
\sigma
\delta
\mu
\right) , (2.3b)
where
\mu :=
\zeta 1 + \zeta 2
2
and \sigma :=
\biggl(
\zeta 1 - \zeta 2
2
\biggr) 2
= \mu 2 - \zeta 1 \zeta 2 . (2.3c)
Note that (2.3a) indeed holds true for arbitrary \delta \not = 0 because
CL =
\Biggl(
0 1
- \zeta 1 \zeta 2 2\mu
\Biggr) \Biggl(
1 0
\mu \delta
\Biggr)
=
\Biggl(
\mu \delta
2\mu 2 - \zeta 1 \zeta 2 2\mu \delta
\Biggr)
,
LT =
\Biggl(
1 0
\mu \delta
\Biggr) \left( \mu \delta
\sigma
\delta
\mu
\right) =
\Biggl(
\mu \delta
\mu 2 + \sigma 2\mu \delta
\Biggr)
= CL.
Optimal choices for the scaling parameter \delta in dependence of \{ \zeta 1, \zeta 2\} will be specified later on.
Remark 2.1 (Vandermonde decompositions). The transformations discussed above are related to
LU- and LQ-decompositions of the Vandermonde matrix V. We have
V =
\Biggl(
1 1
\zeta 1 \zeta 2
\Biggr)
=
\Biggl(
1 0
\zeta 1 1
\Biggr) \Biggl(
1 1
0 \zeta 2 - \zeta 1
\Biggr)
= L \cdot U,
where L is lower diagonal and U is upper diagonal. On the other hand,
V =
\Biggl(
1 1
\zeta 1 \zeta 2
\Biggr)
=
\Biggl(
1 0
\mu 1
\Biggr) \left( 1 1
1
2
(\zeta 1 - \zeta 2)
1
2
(\zeta 2 - \zeta 1)
\right) = L \cdot Q, \mu =
1
2
(\zeta 1 + \zeta 2),
where L is lower diagonal and where the rows of Q are orthogonal to each other.
3. Contractivity for stable spectra in the closed unit circle. Assume that C from (2.1a)
satisfies a stability condition (root condition) with respect to the closed complex unit circle, i.e.,
| \zeta 1| \leq 1, | \zeta 2| \leq 1, and | \zeta 1| < 1 if \zeta 1 = \zeta 2. (3.1)
Proposition 3.1 (similarity to contraction). Consider a companion matrix of the form (2.1a),
C \in \BbbC 2\times 2 with spectrum \{ \zeta 1, \zeta 2\} , satisfying the stability condition (3.1). Let
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
A NOTE ON SIMILARITY TO CONTRACTION FOR STABLE 2\times 2 COMPANION MATRICES 403
\delta =
\sqrt{}
1
2
(1 - | \zeta 1| 2)(1 - | \zeta 2| 2) +
1
4
| \zeta 1 - \zeta 2| 2 > 0. (3.2)
Then the transformed matrix T from (2.3) satisfies
\| T\| 2 \leq 1. (3.3)
The parameter \delta from (3.2) is a measure for “the distance to instability” of the spectrum \{ \zeta 1, \zeta 2\} .
It vanishes exactly in the limiting, unstable case \zeta 1 = \zeta 2 with | \zeta 1| = | \zeta 2| = 1. For further details
concerning this similarity transformation, see Remark 3.1 below.
Proof. We consider (2.3) with the parameter \delta unspecified for the moment. The norm \| T\| 2
cannot be expressed in a simple way as a function of \delta . Alternatively, we aim for finding \delta > 0 such
that the requirement
S := \Delta 2 - (T\Delta )\ast (T\Delta ) \geq [>] 0, with \Delta =
\biggl(
1 0
0 \delta
\biggr)
, (3.4)
is satisfied, which is equivalent to the requirement \| T\| 2 \leq [<] 1.
The matrix S evaluates to
S =
\Biggl(
1 - | \mu | 2 - \mu \=\sigma
- \sigma \=\mu - | \sigma | 2
\Biggr)
+ \delta 2
\Biggl(
- 1 - \=\mu
- \mu 1 - | \mu | 2
\Biggr)
,
and its determinant is given by
\mathrm{d}\mathrm{e}\mathrm{t} S = - \delta 4 +
\bigl(
1 - 2 | \mu | 2 + | \mu 2 - \sigma | 2
\bigr)
\delta 2 - | \sigma | 2. (3.5)
This assumes its maximal value for
\delta 2 =
1
2
\bigl(
1 - 2| \mu | 2 + | \mu 2 - \sigma | 2
\bigr)
=
1
2
(1 - | \zeta 1| 2)(1 - | \zeta 2| 2) + (1 - | \mu | 2) =
=
1
2
(1 - | \zeta 1| 2)(1 - | \zeta 2| 2) +
1
4
| \zeta 1 - \zeta 2| 2 \geq 0. (3.6)
With this choice for \delta > 0, i.e., \delta according to (3.2), \mathrm{d}\mathrm{e}\mathrm{t} S evaluates to
\mathrm{d}\mathrm{e}\mathrm{t}S = \delta 4 - | \sigma | 2 =
\biggl(
\delta 2 - 1
4
| \zeta 1 - \zeta 2| 2
\biggr) \biggl(
\delta 2 +
1
4
| \zeta 1 - \zeta 2| 2
\biggr)
=
=
1
2
(1 - | \zeta 1| 2)(1 - | \zeta 2| 2)
\biggl(
\delta 2 +
1
4
| \zeta 1 - \zeta 2| 2
\biggr)
=
1
4
(1 - | \zeta 1| 2)(1 - | \zeta 2| 2) | 1 - \zeta 1 \=\zeta 2|
2 \geq 0.
Now we check requirement (3.4) for S with \delta 2 from (3). To this end, we note that
trace S =
\bigl(
1 - | \mu | 2 - \delta 2
\bigr)
+
\bigl(
- | \sigma | 2 + \delta 2(1 - | \mu | 2)
\bigr)
,
and
| \mu | 2 + | \sigma | = 1
4
| \zeta 1 + \zeta 2| 2 +
1
4
| \zeta 1 - \zeta 2| 2 =
1
2
\bigl(
| \zeta 1| 2 + | \zeta 2| 2
\bigr)
. (3.7)
We consider three different cases of a stable spectrum (in all cases, | \mu | < 1 and \delta > 0):
(i) | \zeta 1| < 1, | \zeta 2| < 1. Here,
\delta 2 < 1 - | \mu | 2, S11 > 0, \mathrm{d}\mathrm{e}\mathrm{t}S > 0.
This implies S > 0.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
404 W. AUZINGER
(ii) | \zeta 1| = 1, | \zeta 2| < 1. Here,
\delta 2 = 1 - | \mu | 2, \mathrm{d}\mathrm{e}\mathrm{t}S = 0, trace S > 0,
where the estimate for the trace easily follows from (3.7). This implies that the eigenvalues of S
must be \lambda 1 = 0 and \lambda 2 > 0, hence S \geq 0 with rank(S) = 1.
(iii) | \zeta 1| = | \zeta 2| = 1, with \zeta 1 \not = \zeta 2. Here,
\delta 2 = 1 - | \mu | 2, \mathrm{d}\mathrm{e}\mathrm{t}S = 0, trace S = 0,
where traceS = 0 again follows from (3.7). This implies S = 0.
In all these cases, rank(S) equals the number of eigenvalues \zeta k with | \zeta k| < 1. Summarizing
(i)– (iii) concludes the proof.
Remark 3.1 (special cases). For \rho (C) = 1 with S = 0 (case (iii) above), C is diagonalizable,
and S = 0 implies that T is unitary.
For \rho (C) = 1 with 0 \not = S \geq 0 (case (ii) above), C is diagonalizable. In this case it follows from
\delta 2 = | \sigma | that T is normal, with \| T\| 2 = 1. Thus, up to unitary transformation the outcome amounts
to diagonalization of C. We may call T a normalization of C.
In cases (ii) and (iii), \delta =
\sqrt{}
1 - | \mu | 2 is approximately proportional to the distance between 1 and
the modulus of the arithmetic mean \mu of the eigenvalues of the matrix C.
The more interesting, general case is \rho (C) < 1, with S > 0 (case (i) above):
For \zeta 1 \not = \zeta 2, T is not related to a diagonalization, or normalization, of C, which gets undefined
in the limit \zeta 1 \rightarrow \zeta 2. The transformation matrix L is well-conditioned also for \zeta 1 \rightarrow \zeta 2 unless the
spectrum is close to the unit circle. Here we have S > 0 and \| T\| 2 < 1. The value of \| T\| 2 depends
on the location of the spectrum of C in a rather complicated way.
In the confluent case \zeta 1 = \zeta 2 = \mu we obtain \delta =
\surd
2
2
(1 - | \mu | 2), thus
T =
\left( \mu
\surd
2
2
(1 - | \mu | 2)
0 \mu
\right) ,
i.e., T is a rescaled Jordan form.
In all cases, the condition number of the transformation matrix L is \scrO (\delta - 1) for \delta \rightarrow 0, which
is quite natural and related to the transient behavior of the powers \| C\nu \| 2 (the so-called hump
phenomenon).
Summarizing, we see that Proposition 3.1 describes a similarity transformation leading to a
contraction which is based on a smooth transition between normalization and Jordan decomposition.
The construction appears quite natural, but it is not unique, because one may think of different ways
to balance between “\| T\| 2 small” and “\kappa (L) not too large”.
Example 3.1 (Second order difference equations). Consider the homogeneous difference equa-
tion
y\nu +2 + c1 y\nu +1 + c0 y\nu = 0, \nu \geq 0,
for given y0, y1. For the characteristic polynomial \pi (\zeta ) = \zeta 2 + c1\zeta + c0 = (\zeta - \zeta 1)(\zeta - \zeta 2) we
assume that \{ \zeta 1, \zeta 2\} satisfies the stability condition (3.1). With \bfity \nu = (y\nu , y\nu +1)
T this is equivalent to
\bfity \nu +1 = C \bfity \nu with C from (2.1a), or equivalently, L - 1\bfity \nu +1 = T L - 1\bfity \nu with L, T from (2.3b). Here,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
A NOTE ON SIMILARITY TO CONTRACTION FOR STABLE 2\times 2 COMPANION MATRICES 405
L - 1\bfity \nu =
\left( y\nu
1
\delta
(y\nu +1 - \mu y\nu )
\right) ,
and Proposition 3.1 asserts that
\delta 2 \| L - 1\bfity \nu \| 22 = | \delta y\nu | 2 + | y\nu +1 - \mu y\nu | 2
is always monotonously decreasing with \nu .
4. Dissipativity for stable spectra in the closed left half plane. Assume that C from (2.1a)
satisfies a stability condition (root condition) with respect to the closed complex left half plane, i.e.,
Re \zeta 1 \leq 0, Re \zeta 2 \leq 0, and Re \zeta 1 < 0 if \zeta 1 = \zeta 2. (4.1)
Proposition 4.1 (similarity to dissipation). Consider a companion matrix of the form (2.1a), C \in
\in \BbbC 2\times 2 with spectrum \{ \zeta 1, \zeta 2\} , satisfying the stability condition (4.1). Let
\delta =
\sqrt{}
2 Re \zeta 1Re \zeta 2 +
1
4
| \zeta 1 - \zeta 2| 2 > 0. (4.2)
Then the transformed matrix T from (2.3) satisfies
Re T =
1
2
(T + T \ast ) \leq 0. (4.3)
The parameter \delta from (4.2) is a measure for “the distance to instability” of the spectrum \{ \zeta 1, \zeta 2\} . It
vanishes exactly in the limiting, unstable case \zeta 1 = \zeta 2 with Re \zeta 1 = Re \zeta 2 = 0.
Proof. We aim for finding \delta > 0 such that the requirement
S := Re (2\delta T ) \leq [<] 0 (4.4)
is satisfied, which is equivalent to the requirement Re T \leq [<] 0.
The matrix S evaluates to
S =
\Biggl(
2\delta Re \mu \delta 2 + \=\sigma
\delta 2 + \sigma 2\delta Re \mu
\Biggr)
,
and its determinant is given by
\mathrm{d}\mathrm{e}\mathrm{t} S = - \delta 4 + 2
\bigl(
2(Re \mu )2 - Re\sigma
\bigr)
\delta 2 - | \sigma | 2. (4.5)
This assumes its maximal value for
\delta 2 = 2(Re \mu )2 - Re\sigma = 2 Re \zeta 1Re \zeta 2 +
1
4
| \zeta 1 - \zeta 2| 2 \geq 0. (4.6)
With this choice for \delta > 0, i.e., \delta according to (4.2), \mathrm{d}\mathrm{e}\mathrm{t} S evaluates to
\mathrm{d}\mathrm{e}\mathrm{t} S = \delta 4 - | \sigma | 2 =
\biggl(
\delta 2 - 1
4
| \zeta 1 - \zeta 2| 2
\biggr) \biggl(
\delta 2 +
1
4
| \zeta 1 - \zeta 2| 2
\biggr)
=
= 2 Re \zeta 1Re \zeta 2
\bigl(
2 Re \zeta 1Re \zeta 2 +
1
2
| \zeta 1 - \zeta 2| 2
\bigr)
= Re \zeta 1Re \zeta 2 | \zeta 1 + \zeta 2| 2.
Now we check requirement (4.4) for S with \delta 2 from (4.6).
We consider three different cases of a stable spectrum (in all cases, Re \mu < 0 and \delta > 0):
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
406 W. AUZINGER
Fig. 1. Damped harmonic oscillator. Energy E(y(t), \.y(t)) [\diamond \diamond \diamond ] and mean energy \~E(y(t), \.y(t)) [ - - ] for \omega = 1.5, \rho =
= 0.25, and initial values y(0) = \.y(0) = 1.
(i) Re \zeta 1 < 0, Re \zeta 2 < 0. Here,
\delta 2 > | \sigma | , Re \mu < 0, S11 < 0, \mathrm{d}\mathrm{e}\mathrm{t}S > 0.
This implies S < 0.
(ii) Re \zeta 1 = 0, Re \zeta 2 < 0. Here,
\delta 2 = | \sigma | , Re \mu < 0, \mathrm{d}\mathrm{e}\mathrm{t}S = 0, trace S < 0.
This implies that the eigenvalues of S must be \lambda 1 = 0 and \lambda 2 < 0, hence S \leq 0 with rank(S) = 1.
(iii) Re \zeta 1 = Re \zeta 2 = 0, with \zeta 1 \not = \zeta 2. Here,
\delta 2 = | \sigma | , Re \mu = 0, \mathrm{d}\mathrm{e}\mathrm{t}S = 0, trace S = 0.
This implies S = 0.
In all these cases, rank(S) equals the number of eigenvalues \zeta k with Re \zeta k < 0. Summarizing
(i)– (iii) concludes the proof.
Remark 4.1 (special cases). Similar remarks as those following Proposition 3.1 apply. For case
(iii), in particular, S = 0 implies Re T = 0, i.e., T is skew-Hermitian. For case (ii), T is normal.
Example 4.1 (damped harmonic oscillator). In this example we show that, in the context of a
simple ODE problem, Proposition 4.1 provides a physically meaningful dissipation functional.
Consider the second order linear ODE for the free damped harmonic oscillator in the dimensionless
variable y,
\"y(t) + 2\rho \.y(t) + \omega 2 y(t) = 0,
with damping parameter \rho \geq 0 and angular frequency \omega > 0. For \bfity (t) = (y(t), \.y(t))T we have
\.\bfity (t) = C \bfity (t), C =
\Biggl(
0 1
- \omega 2 - 2\rho
\Biggr)
,
with eigenvalues \zeta 1,2 = - \rho \pm
\sqrt{}
\rho 2 - \omega 2 and \mu =
1
2
(\zeta 1 + \zeta 2) = - \rho . Consider the assertion from
Proposition 4.1. In all cases (over- or underdamping, critical damping) for \delta from (4.2) we obtain
\delta =
\sqrt{}
\rho 2 + \omega 2, and
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
A NOTE ON SIMILARITY TO CONTRACTION FOR STABLE 2\times 2 COMPANION MATRICES 407
T =
\left( - \rho
\sqrt{}
\rho 2 + \omega 2
\rho 2 - \omega 2\sqrt{}
\rho 2 + \omega 2
- \rho
\right) with Re T \leq
\Bigl( \rho \sqrt{}
\rho 2 + \omega 2
- 1
\Bigr)
\rho I \leq 0.
Together with
d
dt
(L - 1\bfity (t)) = T (L - 1\bfity (t)) this implies
\| L - 1\bfity (t)\| 2 \leq e - \~\rho t \| L - 1\bfity (0)\| 2, with \~\rho := - \mu 2(T ) =
\Bigl(
1 - \rho \sqrt{}
\rho 2 + \omega 2
\Bigr)
\rho \geq 0.
(Here, \mu 2(T ) denotes the logarithmic norm of T, i.e. the rightmost eigenvalue of Re T.) Equivalently,
this means that
\~E(y, \.y) := (\rho 2 + \omega 2)\| L - 1\bfity \| 22 = (\rho 2 + \omega 2)y2 + ( \.y + \rho y)2
is always a Lyapunov functional for the oscillator, i.e., d \~E \leq 0 along solution trajectories. In the un-
damped case, \~E is identical with the total energy functional E(y, \.y) = \omega 2 y2+ \.y2 which is conserved,
d \~E \equiv 0 for \rho = 0. For \rho > 0 we have dE < 0, and d \~E < 0 due to \~\rho > 0, where \~E \not = E. A straight-
forward calculation shows d \~E = - 2\rho E, i.e., \~E(t) represents a form of mean energy (see Fig. 1).
A remarkable special case occurs, e.g., at confluence, \omega = \rho \gg 0, critical damping at high
stiffness. Here,
\mu 2(C) =
1
2
(\omega - 1)2 = \scrO (\omega 2) \gg 0, in contrast to \mu 2(T ) =
\Biggl( \surd
2
2
- 1
\Biggr)
\omega \ll 0.
References
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Received 07.06.13,
after revision — 30.11.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 3
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| id | umjimathkievua-article-1847 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:48Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c6/494d1922c81be1661d4761a70b3ab3c6.pdf |
| spelling | umjimathkievua-article-18472019-12-05T09:29:34Z A note on similarity to contraction for stable $2 \times 2$ companion matrices Про подiбнiсть вiдносно стиску для стабiльних $2 \times 2$ супутнiх матриць Auzinger, W. Аузінгер, В. We consider companion matrices of size $2 \times 2$ with general complex spectra satisfying a root condition with respect to the closed complex unit circle or the closed left complex half plane. For both cases, smooth and naturally conditioned basis transformations are constructed such that the resulting, transformed matrix is contractive or dissipative, respectively, with respect to the $\ell_2$-norm. Розглядаються супутнi матрицi розмiром $2 \times 2$, загальнi комплекснi спектри яких задовольняють кореневу умову вiдносно замкненого комплексного одиничного кола або замкненої лiвої комплексної напiвплощини. В обох ви- падках будуються гладкi базиснi природно обумовленi перетворення такi, що результуюча перетворена матриця є стискаючою або дисипативною вiдносно $\ell_2$-норми, вiдповiдно. Institute of Mathematics, NAS of Ukraine 2016-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1847 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 3 (2016); 400-407 Український математичний журнал; Том 68 № 3 (2016); 400-407 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1847/829 Copyright (c) 2016 Auzinger W. |
| spellingShingle | Auzinger, W. Аузінгер, В. A note on similarity to contraction for stable $2 \times 2$ companion matrices |
| title | A note on similarity to contraction for stable $2 \times 2$ companion matrices |
| title_alt | Про подiбнiсть вiдносно стиску для стабiльних $2 \times 2$ супутнiх матриць |
| title_full | A note on similarity to contraction for stable $2 \times 2$ companion matrices |
| title_fullStr | A note on similarity to contraction for stable $2 \times 2$ companion matrices |
| title_full_unstemmed | A note on similarity to contraction for stable $2 \times 2$ companion matrices |
| title_short | A note on similarity to contraction for stable $2 \times 2$ companion matrices |
| title_sort | note on similarity to contraction for stable $2 \times 2$ companion matrices |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1847 |
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