**Theorem 3.1.1.**

Let *J* be an open interval in ℝ, and let *λ*) ∈ ℂ*nxn*, *λ* ∈ J, be a family of Hermitian matrices depending continuously on the parameter *λ* ∈ *J*, such that the conditions (A) and (B) are satisfied. Then the following statements hold.

**a.** For every *l* ∈ ℕ there is at most one *lth* eigenvalue of *T*(⋅) which can be characterized by

$$\mathcal{A}\_I = \min\_{V \in H\_I, V \subset D} \sup\_{\nu \in V \subset D} p(\nu) \tag{13}$$

**b.** If

$$\mathcal{A}\_I \coloneqq \inf\_{V \in H\_I, V \cap D \neq \mathcal{Q}} \sup\_{\mathbf{v} \in V \cap D} p(\mathbf{v}) \in J.$$

For some *l* ∈ ℕ then *λ <sup>l</sup>* is the lth eigenvalue of ,*T*(⋅) in *J*, and (13) holds.


$$\mathcal{A}\_l = \min\_{V \in H\_l, V \supset D \cup \{0\}} \sup\_{\nu \in V, \nu \neq 0} p(\nu).$$


Sylvester's law of inertia has an important role in the nonlinear eigenvalue problems. We will briefly look back to the Sylvester's law of inertia. With this purpose we define the inertia of the Hermitian matrix *T* as follows [9]**.**

**Definition 3.1.1.** The inertia of a Hermitian matrix *T* is the triplet of nonnegative integers In(*T*) = (*np*, *nn*, *nz*) where *np*, *nn* and *nz* are the number of positive, negative, and zero eigen‐ values of *T* (counting multiplicities).

Next, we consider a case that an extreme eigenvalue *λ*1 := *infx* <sup>∈</sup> *Dp*(*x*)or*λn* := *supx* <sup>∈</sup> *Dp*(*x*) is contained in *J*.

**Theorem 3.1.2** Assume that *T* : *J* → ℂ*nxn* satisfies the conditions of the minmax characterization, and let (*np*, *nn*, *nz*) be the inertia of *T*(*σ*) for some *σ* ∈ *J*.


We consider the quadratic eigenvalue problem in the label QEP .That is why we adapt to the real scalar equation (11) QEPu. In this way, we get

$$\mathbf{f}(\lambda; \mathbf{x}; \mathbf{=}) \lambda^2 \mathbf{x}^H A \mathbf{x} + \lambda \mathbf{x}^H B \mathbf{x} + \mathbf{x}^H \mathbf{C} \mathbf{x} \mathbf{=} \mathbf{0} \quad \text{for every fixed } \mathbf{x} \in \mathbb{C}^n, \mathbf{x} \neq \mathbf{0} \tag{14}$$

Natural candidates for the Rayleigh functionals of QEPa (**Eq. 10**) are

(B) for every *x* ∈ *D* and every *λ* ∈ *J* with *λ* ≠ *p*(*x*) it holds that (*λ* − *p*(*x*))*f*(*λ*; *x*) > 0.

**nonoverdamped**.

74 Applied Linear Algebra in Action

**Theorem 3.1.1.**

**b.** If

terization for eigenvalues.

For some *l* ∈ ℕ then *λ <sup>l</sup>*

**d.** Let *λ* 1 = *infx* ∈ *Dp*(*x*) ∈ *J* and *λ <sup>l</sup>*

eigenvalue *λ <sup>j</sup>*

*T*(*λ*)**x** = *μ***x**.

largest eigenvalues.

the Hermitian matrix *T* as follows [9]**.**

satisfied. Then the following statements hold.

If *p* is defined on *D* = ℂ*<sup>n</sup>*\{**0**} then the problem (1) is called **overdamped**, otherwise it is called

Generalizations of the minmax and the maxmin characterizations of the eigenvalues were proved by Duffin [6] for the quadratic case and by Rogers [7] for the general overdamped problems. For the nonoverdamped eigenproblems the natural ordering to call the smallest eigenvalue the first one, the second smallest the second one, etc., is not appropriate. The next theorem is proved in [8], which gives more information about the following minmax charac‐

Let *J* be an open interval in ℝ, and let *λ*) ∈ ℂ*nxn*, *λ* ∈ J, be a family of Hermitian matrices depending continuously on the parameter *λ* ∈ *J*, such that the conditions (A) and (B) are

**a.** For every *l* ∈ ℕ there is at most one *lth* eigenvalue of *T*(⋅) which can be characterized by

( ) , min sup

( ) , : inf sup .

> ( ) , {0} , 0 min sup .

 *pv J* Î Ç ¹Æ Î Ç = Î

is the lth eigenvalue of ,*T*(⋅) in *J*, and (13) holds.

 ≤ *λ <sup>l</sup>* .

*p v* Î ÉÈ Î ¹

<sup>=</sup> (13)

∈ *J*. If the minimum in (13) is attained for an *l* dimensional

in *J*(*k* < *l*), then *J* contains the *jth*

) corresponding to its *lth*

*l*

l

l

**c.** If there exists the *kth* and *lth* eigenvalue *λ <sup>k</sup>* and *λ <sup>l</sup>*

(*k* ≤ *j* ≤ *l*) as well, and *λ <sup>k</sup>* ≤ *λ <sup>j</sup>*

subspace *V*, then *V* ⊃ *D* ∪ {0}, and (3) can be replaced with

l

**f.** The minimum in (3) is attained for the invariant subspace of *T*(*λ<sup>l</sup>*

=

*<sup>l</sup> V HV D vV D*

*p v* Î Ç ¹Æ Î Ç

*l <sup>l</sup> V HV D vV D*

*l <sup>l</sup> V HV D vVv*

**e.** *λ*˜ is an *lth* eigenvalue if and only if *μ* = 0 is the lth eigenvalue of the linear eigenproblem

Sylvester's law of inertia has an important role in the nonlinear eigenvalue problems. We will briefly look back to the Sylvester's law of inertia. With this purpose we define the inertia of

$$p\_+\left(\mathbf{x}\right) \coloneqq -\frac{\mathbf{x}^H \, \mathbf{B} \mathbf{x}}{2\mathbf{x}^H A \mathbf{x}} + \sqrt{\left(\frac{\mathbf{x}^H \, \mathbf{B} \mathbf{x}}{2\mathbf{x}^H A \mathbf{x}}\right)^2 - \frac{\mathbf{x}^H \mathbf{C} \mathbf{x}}{\mathbf{x}^H A \mathbf{x}}} \qquad\text{and}\tag{15}$$

$$p\_{-}(\mathbf{x}) \coloneqq -\frac{\mathbf{x}^{H}\,\mathbf{B}\mathbf{x}}{2\mathbf{x}^{H}\,A\mathbf{x}} - \sqrt{\left(\frac{\mathbf{x}^{H}\,\mathbf{B}\mathbf{x}}{2\mathbf{x}^{H}\,A\mathbf{x}}\right)^{2} - \frac{\mathbf{x}^{H}\,\mathbf{C}\mathbf{x}}{\mathbf{x}^{H}\,A\mathbf{x}}}\tag{16}$$

The Rayleigh functionals are the generalization of the Rayleigh quotient.

In this section we deal with the hyperbolic quadratic pencil as an example overdamped problems and gyroscopically stabilized pencil as an example of the changes that are not.

Now, let us look briefly the hyperbolic quadratic pencil. It is overdamped square pencil given by (10) in which the *A* = **A**H > 0, *B* = *B<sup>H</sup>*, *C* = **C***<sup>H</sup>***.** For hyperbolic quadratic pencil, the following interesting features

The ranges *J*+ := *p*+(ℂ*<sup>n</sup>*\{**0**}) and *J*− := *p*−(ℂ*<sup>n</sup>*\{**0**}) are disjoint real intervals with max *J*− < min *J*+. **Q**(*λ*) is the positive definite for *λ* < min *J*− and *λ* > max *J*+, and it is the negative definite for *λ* ∈ (max *J*−, min *J*+).

(*Q*, *J*+) and (−*Q*, *J*−) satisfy the conditions of the variational characterization of the eigenvalues, i.e. there exist 2*n* eigenvalues[1].

$$
\lambda\_1 \le \lambda\_2 \le \cdots \le \lambda\_n < \lambda\_{n+1} \le \cdots \le \lambda\_{2n} \tag{17}
$$

and

$$\mathcal{A}\_{j} = \min\_{\dim V = j} \max\_{\mathbf{x} \in V, \ \mathbf{x} \neq \mathbf{0}} p\_{-}(\mathbf{x}), \quad \mathcal{A}\_{n+j} = \min\_{\dim V = j} \max\_{\mathbf{x} \in V, \mathbf{x} \neq \mathbf{0}} p\_{+}(\mathbf{x}), \quad j = 1, 2, \dots, n. \tag{18}$$

Now we will look at gyroscopically stabilized system in the label GSS. A quadratic polynomial matrix

$$\underline{\boldsymbol{Q}}(\boldsymbol{\lambda}) = \boldsymbol{\lambda}^2 \boldsymbol{I} + \boldsymbol{\lambda}\boldsymbol{\mathcal{B}} + \boldsymbol{C}, \quad \underline{\mathbf{B}} = \mathbf{B}^H, \text{ det}(\boldsymbol{\mathcal{B}}) \neq \mathbf{0}, \ \mathbf{C} = \mathbf{C}^H \tag{19}$$

is gyroscopically stabilized if for some *k>0* it holds that

$$
\left| \mathbf{B} \right| > kI + k^{-1}\mathbf{C}, \tag{20}
$$

where denotes the positive square root of *B*<sup>2</sup> .

**Definition 3.1.2**. A eigenvalue λ is positive type if applies **xH***Q ′* (*λ*)**x**>0 **x**∈ℂ*n*, **x**≠**0**

A eigenvalue λ is positive type if applies **xH***Q ′* (*λ*)**x**<0 **x**∈ℂ*n*, **x**≠**0**

Theorem (Barkwell, Lancaster, Markus 1992)


Without loss of generality we will observe only positive eigenvalues value

$$\begin{aligned} \text{Let } & \text{let now } p\_{+}(\mathbf{x}) \coloneqq -\frac{\mathbf{x}^{\text{il}} \cdot \mathbf{x} \mathbf{x}}{2\mathbf{x}^{\text{il}}\mathbf{x}} + \sqrt{\left(\frac{\mathbf{x}^{\text{il}} \cdot \mathbf{x}\mathbf{x}}{2\mathbf{x}^{\text{il}}\mathbf{x}}\right)^{2} - \frac{\mathbf{x}^{\text{il}} \cdot \mathbf{x}}{\mathbf{x}^{\text{il}}\mathbf{x}}} \text{ and } p\_{-}(\mathbf{x}) \coloneqq -\frac{\mathbf{x}^{\text{il}} \cdot \mathbf{x}}{2\mathbf{x}^{\text{il}}\mathbf{x}} - \sqrt{\left(\frac{\mathbf{x}^{\text{il}} \cdot \mathbf{x}}{2\mathbf{x}^{\text{il}}\mathbf{x}}\right)^{2} - \frac{\mathbf{x}^{\text{il}} \cdot \mathbf{x}}{\mathbf{x}^{\text{il}}\mathbf{x}}} \text{ functionals approach} \\\\ \text{partial } p\_{-}(\mathbf{x}) \text{ for GSS. With them, we can define the Rayleigh functionals } p\_{-}^{+} \coloneqq \begin{vmatrix} p\_{-}(\mathbf{x}) & \text{za} & p\_{-}(\mathbf{x}) > 0 \\ & 0 & \text{else} \end{vmatrix} \\\\ p\_{+}^{+} \coloneqq \begin{vmatrix} p\_{+}(\mathbf{x}) & \text{za} & p\_{-}(\mathbf{x}) > 0 \\ & 0 & \text{else} \end{vmatrix} \end{aligned}$$

Voss and Kostić are defined for this function given interval in which the eigenvalues can minmax characterize.

In order to minmax characterized all eigenvalues, we will introduce new Rayleigh functional. It is a new strategy. With this aim we matrices *B* and *C* as well as vector **x** present in the following format :

$$\mathbf{B} = \begin{pmatrix} \mathbf{B}\_1 & \mathbf{0} \\ 0 & -\mathbf{B}\_2 \end{pmatrix}, \mathbf{B}\_i > 0, \begin{pmatrix} i = 1, 2 \end{pmatrix}$$

$$\mathbf{C} = \begin{pmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} \\ \mathbf{C}\_{12}^H & \mathbf{C}\_{22} \end{pmatrix}, \mathbf{C}\_{ii} > 0, \begin{pmatrix} i = 1, 2 \end{pmatrix}$$

$$\mathbf{x} = \begin{pmatrix} \mathbf{z} \\ \mathbf{y} \end{pmatrix} \neq \mathbf{0}.$$

We define

1 2 £ ££ < ££

( ) ( ) dim , 0 dim , 0

 l

 L L l

min max , min max , 1,2, , . *<sup>j</sup> n j V j xV x V j xVx*

<sup>2</sup> ( )= + = ¹= , , det( ) 0, *<sup>H</sup> Q C*

.

**a.** The spectrum of a gyroscopic stabilized pencil is real, i.e. Q is quasi-hyperbolic.

**c.** If (*np*, *nn*, *nz*) is the inertia of *B*, then *Q*(*λ*)x = **0** has 2*np* negative and 2 *nn* positive eigen‐

**d.** The 2*np* negative eigenvalues lie in two disjoint intervals, eigenvalues in each; the ones in the left interval are of negative type, the ones in the right interval are of positive type.

**e.** The 2 *nn* negative eigenvalues lie in two disjoint intervals, eigenvalues in each; the ones in the left interval are of negative type, the ones in the right interval are of positive type.

*<sup>p</sup> pj n* - + <sup>+</sup> = = Î ¹ Î ¹

Now we will look at gyroscopically stabilized system in the label GSS. A quadratic polynomial

ll

*nn n* l

+1 2

= == **x x** <sup>L</sup> (18)

(17)

*<sup>H</sup> I+ B B B B C C* (19)

<sup>1</sup> *k k* , - *BI C* > + (20)

(*λ*)**x**<0 **x**∈ℂ*n*, **x**≠**0**

(*λ*)**x**>0 **x**∈ℂ*n*, **x**≠**0**

l

ll

where denotes the positive square root of *B*<sup>2</sup>

A eigenvalue λ is positive type if applies **xH***Q ′*

Theorem (Barkwell, Lancaster, Markus 1992)

values.

is gyroscopically stabilized if for some *k>0* it holds that

 l

**Definition 3.1.2**. A eigenvalue λ is positive type if applies **xH***Q ′*

**b.** All eigenvalues are either of positive type or of negative type

Without loss of generality we will observe only positive eigenvalues value

and

matrix

l

76 Applied Linear Algebra in Action

$$\begin{aligned} \mathcal{Q}\_{\mathbb{I}}\left(\boldsymbol{\lambda}\right) & \coloneqq \boldsymbol{\lambda}^2 \boldsymbol{I} + \boldsymbol{\lambda}\boldsymbol{\mathcal{B}}\_{\mathbb{I}} + \boldsymbol{\mathcal{C}}\_{\text{11}} \\ \mathcal{Q}\_{\mathbb{2}}\left(\boldsymbol{\lambda}\right) & \coloneqq \boldsymbol{\lambda}^2 \boldsymbol{I} - \boldsymbol{\lambda}\boldsymbol{\mathcal{B}}\_{\mathbb{2}} + \boldsymbol{\mathcal{C}}\_{\text{22}} \\ \mathcal{T}\left(\boldsymbol{\lambda}\right) & \coloneqq \boldsymbol{\mathcal{Q}}\_{\mathbb{2}}\left(\boldsymbol{\lambda}\right) - \boldsymbol{\mathcal{C}}\_{\text{12}}^H \left(\boldsymbol{\mathcal{Q}}\_{\mathbb{I}}\left(\boldsymbol{\lambda}\right)\right)^{-1} \boldsymbol{\mathcal{C}}\_{\text{12}} \end{aligned}$$

Because of the conditions (20) are *Q***1**(*λ*) **I** *Q***2**(*λ*) are hyperbolic. We will observe the following problem inherent value

$$T(\mathcal{A})\,\,\mathbf{y} = 0, \,\mathbf{y} \neq 0, \mathbf{y} \in \mathbf{C}^{n-n\_\rho}, \mathbf{y} \neq \mathbf{0}$$

Applies following theorem

**Theorem 3.1.3** λ is eigenvalue *Q*(⋅) if and only if λ is eigenvalue *T*(⋅)

**Proof**

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ) ( ) ( ) () () ( ) ( ) ( ) ( ) 1 12 1 12 12 2 12 2 1 12 1 1 12 12 2 12 2 12 2 1 1 1 2 12 1 12 1 1 0 . *H H H H* l l l l l l l l l l l ll l - - æ öæ öæ ö <sup>=</sup> Û = ç ÷ç ÷ Û = ç ÷ è ø è øè ø æ ö + æ ö Û ç ÷ = Û + =Ù + = ç ÷ + è ø è ø Û =- Ù æ ö æ ö ç ÷ <sup>=</sup> ç ÷ è ø è ø Û =- æ ö ç ÷ Ù = è ø **<sup>z</sup> x 0 x 0 <sup>0</sup> y 0 0 z y0 z y 0 y 0 y** *H QC QC Q CQ CQ Q zCy Q zCy C Q C Q z Q Cy Q C Q C z Q yC T*

Theorem is proved.

Analogous to the (14) we defined the following functions

$$q(\boldsymbol{\lambda}; \mathbf{y}) \coloneqq \mathbf{y}^{\mathbf{H}} T(\boldsymbol{\lambda}) \mathbf{y} \quad \text{and} \quad f\_2(\boldsymbol{\lambda}; \mathbf{y}) \coloneqq \mathbf{y}^{\mathbf{H}} \mathbf{Q}\_2(\boldsymbol{\lambda}) \mathbf{y}$$

In the following theorem we give information about the properties *q*(*λ*; **y**)

**Theorem 3.1.4.**Function *q*(*λ*; **y**) has the following characteristics


#### **Proof**

**a.** Because *C* > 0 we have

$$
\begin{pmatrix}
\end{pmatrix} \cdot \begin{pmatrix}
\mathbf{C}\_{11} & \mathbf{C}\_{12} \\
\mathbf{C}\_{12}^H & \mathbf{C}\_{22}
\end{pmatrix} \cdot \begin{pmatrix}
\mathbf{y}
\end{pmatrix} > 0.
$$

It follows

**Theorem 3.1.3** λ is eigenvalue *Q*(⋅) if and only if λ is eigenvalue *T*(⋅)

l

**0**

( )

l

*Q zCy Q zCy C Q C Q*

*H H*

l

Û ç ÷ = Û + =Ù + = ç ÷

( ) ( ) () () ( )

*z Q Cy Q C Q C*

12

In the following theorem we give information about the properties *q*(*λ*; **y**)

**Theorem 3.1.4.**Function *q*(*λ*; **y**) has the following characteristics

*q*(*λ*; **y**) is greater than the maximum zero function *f*2(*λ*; **y**)

12 11

(*λ*;**y**)>0 *iλ* >0 follow *q*′(*λ*; **y**)

1

l

*z Q yC T*

( ) ( ) ( )

1

*q f* ( ) () ; := = and ; 2 2 ( ) () : llll**H H yy y yy y** *T Q*

**b.** For λ>0 a function *q*(*λ*; **y**) exactly two zeros for each vector y. Minimum zero function *q*(*λ*; **y**) and is lower than the minimum zero function *f*2(*λ*; **y**) and the largest zero function

( ( ) ) ( ) <sup>1</sup>

**<sup>y</sup> y y**

*C C C C C C C C*

<sup>1</sup> 11 12 11 12

è ø è ø

**y**

12 22 0. *H H <sup>H</sup> H* - - æ ö æ ö - - ×× > ç ÷ ç ÷


12

( )

 l

l

**y**

2 12 2

*H*

0 .

( ) ( ( ) ( ) )

ll

*H*

1 12 1 12 12 2 12 2

æ öæ öæ ö <sup>=</sup> Û = ç ÷ç ÷ Û = ç ÷ è ø è øè ø

*QC QC*

*CQ CQ*

**<sup>z</sup> x 0 x 0 <sup>0</sup>**

1 1

1 1 1 2 12 1 12


æ ö æ ö ç ÷ = ç ÷ è ø è ø Û =- æ ö ç ÷ Ù = è ø

( )

**0 z y0**

 l

**y 0**

**y**

 l

( ) ( )

æ ö + æ ö

+ è ø è ø

l

**z y 0**

Analogous to the (14) we defined the following functions

l

Û =- Ù -

( )

l

*H*

l

*Q*

Theorem is proved.

**a.** For each vector *q*(0; **y**) > 0

′

**a.** Because *C* > 0 we have

**c.** From *<sup>f</sup>* <sup>2</sup>

**Proof**

12 2

1 12

**Proof**

78 Applied Linear Algebra in Action

$$\left(\begin{array}{cc} \mathbf{y}^H \mathbf{C}\_{12}^H \left(\mathbf{C}\_{11}\right)^{-1} & \mathbf{y}^H \end{array}\right) \cdot \begin{pmatrix} \mathbf{0} \\ \mathbf{C}\_{22} \mathbf{y} - \mathbf{C}\_{12}^H \left(\mathbf{C}\_{11}\right)^{-1} \mathbf{C}\_{12} \mathbf{y} \end{pmatrix} > 0.1$$

then

$$q\left(0, \mathbf{y}\right) = \mathbf{y}^H \left(\mathbf{C}\_{22} - \mathbf{C}\_{12}^H \left(\mathbf{C}\_{11}\right)^{-1} \mathbf{C}\_{12}\right) \mathbf{y} > 0$$

**b.** We have already mentioned that *Q*2(*λ*) is hyperbolic. This means that the function *f*2(*λ*; **y**) for each vector **y** has two different real roots. For *λ* > 0 ist *Q*1(*λ*) > 0. Therefore, applies

$$
\mathbf{q}\left(\lambda; \mathbf{y}\right) < \mathbf{y}^H \mathbf{Q}\_2\left(\lambda\right) \mathbf{y} - \underbrace{\mathbf{y}^H \mathbf{C}\_{12}^H \left(\mathbf{Q}\_1\left(\lambda\right)\right)^{-1} \mathbf{C}\_{12} \mathbf{y}}\_{>0}
$$

$$
< \mathbf{y}^H \mathbf{Q}\_2\left(\lambda\right) \mathbf{y} = f\_2\left(\lambda; \mathbf{y}\right) \tag{21}
$$

Of further

$$\lim\_{\lambda \to +\infty} q(\lambda; \mathbf{y}) = +\infty \tag{22}$$

Of (a) and because (21) and (22) follows that the function *q*(*λ*; **y**) for λ>0 has two zero. Minimum zero function *q*(*λ*; **y**) *a* and is lower than the minimum zero function *f*2(*λ*; **y**) and the largest zero function *q*(*λ*; **y**) *b* is greater than the maximum zero function *f*2(*λ*; **y**)

$$\mathbf{c}\mathbf{c} \quad q^{'}(\lambda; y) = 2\lambda \, y^{\, H} \, y - y^{\, H} \, B\_{2}y + \, y^{\, H} \, \mathbf{C}\_{12}^{H} \{Q\_{1}(\lambda)\}^{-1} \mathbf{C}\_{12}y$$

Theorem is proved.

Define a new functional

**Definition 3.1.3.** Let *q*(*a*, *y*) = *q*(*b*, *y*) = 0 *and* 0 < *a* < *b*. We define two new functional

$$\begin{aligned} t\_-\left(\mathcal{V}\right) &:= a\\ t\_+\left(\mathcal{V}\right) &:= b\\ W\_\pm &:= t\_\pm\left(h\right) \end{aligned}$$

**Theorem 3.1.5**. Applies *maxW*− < *minW*<sup>+</sup>

**Proof**

$$p\_{2\pm}(\mathbf{x}) \coloneqq -\frac{\mathbf{x}^H \mathbf{B}\_2 \mathbf{x}}{2\mathbf{x}^H \mathbf{x}} \pm \sqrt{\frac{\mathbf{x}^H \mathbf{B}\_2 \mathbf{x}}{2\mathbf{x}^H \mathbf{x}}} \left( \frac{\mathbf{x}^H \mathbf{B}\_2 \mathbf{x}}{2\mathbf{x}^H \mathbf{x}} \right)^2 - \frac{\mathbf{x}^H \mathbf{C}\_2 \mathbf{x}}{\mathbf{x}^H \mathbf{x}}.$$

$$J\_{\pm} \coloneqq p\_{2\pm} \left( \mathbf{C}^{n-n\_{\rho}} \vee \{\mathbf{0}\} \right).$$

$$\max J\_- < \min J\_{\pm}.$$

Because *Q***2**(*λ*) is hyperbolic.

( ) ( ),for every \{ } *<sup>p</sup> n n t y maxJ minJ t y* - - - ++ < << Î**y 0** C

Þ < << *maxW maxJ minJ maxW* - -+ +

Theorem is proved.

**Theorem 3.1.6.** All the positive eigenvalues of (19) are either the value of the MinMax of *t*−(*y*) or Maxmin of *t*+(*y*)

#### **3.2. Linearization**

In this section we will deal with linearization. As mentioned linearization is standard proce‐ dure for reducing QEP on GEP with a view to facilitate the computation of eigenvalues. We have already seen that the problem of eigenvalues usually come as a result of solving differ‐ ential equations or systems of differential equations. That is the basic idea of linearization came in the field of differential equations where the order of the differential equation of the second order can be lowered by introducing a system of two partial differential equations of the first order with two unknown functions.

The basic idea of linearization in QEPa is the introduction of shift **z** = *λ***x** u

$$(\lambda^2 A + \lambda B + C)\_X = 0.$$

Then we get

$$\mathbf{a}. \quad \lambda \mathbf{A} \mathbf{z} + \mathbf{B} \mathbf{z} + \mathbf{C} \mathbf{x} = \mathbf{0} \text{ or } \mathbf{0}$$

**b.** λ*A***z** + λ*B***x** + C**x** = **0**.

The resulting equations are GEP because they can be written respectively in the form of

**a.** ( <sup>−</sup> *<sup>B</sup>* <sup>−</sup>*<sup>C</sup> I O* )(**<sup>z</sup> x** ) <sup>=</sup>*λ*(*A O O I* )(**<sup>z</sup> x** )

**Theorem 3.1.5**. Applies *maxW*− < *minW*<sup>+</sup>

80 Applied Linear Algebra in Action

Because *Q***2**(*λ*) is hyperbolic.

Theorem is proved.

or Maxmin of *t*+(*y*)

**3.2. Linearization**

Then we get

**a.** λ*A***z** + *B***z** + C**x** = **0** or **b.** λ*A***z** + λ*B***x** + C**x** = **0**.

order with two unknown functions.

( )

:

2

±

*p*

( { })

C **0**

*H HH H HH*

æ ö

*B BC*

ç ÷ è ø

\ .: *<sup>p</sup>*

*maxJ minJ*

<

( ) ( ),for every \{ } *<sup>p</sup> n n t y maxJ minJ t y* - - - ++ < << Î**y 0** C

Þ < << *maxW maxJ minJ maxW* - -+ +

**Theorem 3.1.6.** All the positive eigenvalues of (19) are either the value of the MinMax of *t*−(*y*)

In this section we will deal with linearization. As mentioned linearization is standard proce‐ dure for reducing QEP on GEP with a view to facilitate the computation of eigenvalues. We have already seen that the problem of eigenvalues usually come as a result of solving differ‐ ential equations or systems of differential equations. That is the basic idea of linearization came in the field of differential equations where the order of the differential equation of the second order can be lowered by introducing a system of two partial differential equations of the first

*A* + *λB* + *C*)*x* =0.

The resulting equations are GEP because they can be written respectively in the form of

The basic idea of linearization in QEPa is the introduction of shift **z** = *λ***x** u

(*λ* <sup>2</sup>

*n n*



**xx xx xx <sup>x</sup> xx xx xx**

,

2

*J p*

=

± ±

2 2

=- ± - ç ÷

2 2 22

**Proof**

**b.** ( *O C I O*)(**<sup>z</sup> x** ) <sup>=</sup>*λ*(*A B O I* )(**<sup>z</sup> x** ).

Since the corresponding GEPs all matrices 2*n* × 2*n* to GEP has 2n eigenvalue and therefore appropriate QEP has also 2n eigenvalue. From the above it is clear that linearization is not unambiguous. However, in the choice of linearization QEP is an important factor to maintain symmetry and some spectral properties QEP if it is possible. The application of linearization will be will in the next chapter.
