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Let

• ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.

• ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial.

• ${\rm{Q(}}\lambda {\rm{) = }}{{\rm{w}}_m}{\lambda ^m} + .....{w_1}\lambda + {w_0}$

• t=$Q{(\left| \lambda \right|)^2}$

Why does $$\det (tI - P{(\lambda )^*}P(\lambda )) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$$ where $p(x, y)$ and $q(x, y)$ are real polynomials in $x, y$?

Furthermore, if $Q(x)$ is even function, then $p(x,y)=0 $?

Let

• ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.

• ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial.

• ${\rm{Q(}}\lambda {\rm{) = }}{{\rm{w}}_m}{\lambda ^m} + .....{w_1}\lambda + {w_0}$

• t=$Q{(\left| \lambda \right|)^2}$

Why does $$D(x,y)=\det (tI - P{(\lambda )^*}P(\lambda )) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$$ where $p(x, y)$ and $q(x, y)$ are real polynomials in $x, y$?

Furthermore, if $Q(x)$ is even function, then $p(x,y)=0 $?

Let

• ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.

• ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial.

• ${\rm{Q(}}\lambda {\rm{) = }}{{\rm{w}}_m}{\lambda ^m} + .....{w_1}\lambda + {w_0}$

• t=$Q{(\left| \lambda \right|)^2}$

Why does $$\det (tI - P{(\lambda )^*}P(\lambda )) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$$ where $p(x, y)$ and $q(x, y)$ are real polynomials in $x, y$?

Let

• ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.

• ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial.

• ${\rm{Q(}}\lambda {\rm{) = }}{{\rm{w}}_m}{\lambda ^m} + .....{w_1}\lambda + {w_0}$

• t=$Q{(\left| \lambda \right|)^2}$

Why does $$\det (tI - P{(\lambda )^*}P(\lambda )) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$$ where $p(x, y)$ and $q(x, y)$ are real polynomials in $x, y$?

Furthermore, if $Q(x)$ is even function, then $p(x,y)=0 $?

Let

• ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.

• ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial.

• ${\rm{q(}}\lambda {\rm{) = }}{{\rm{w}}_m}{\lambda ^m} + .....{w_1}\lambda + {w_0}$

• t=$q{(\left| \lambda \right|)^2}$

Why does $$\det (tI - P{(\lambda )^*}P(\lambda )) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$$ where $p(x, y)$ and $q(x, y)$ are real polynomials in $x, y\in \mathbb{R}$?

Let

• ${A_j} \in {\mathbb{C}^{n \times n}},0<{w_j}\in \mathbb{R} (j = 0,1,2....m)$ and $\lambda $ is a complex variable such that $\lambda=x+iy$ and $x,y\in \mathbb{R}$.

• ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial.

• ${\rm{Q(}}\lambda {\rm{) = }}{{\rm{w}}_m}{\lambda ^m} + .....{w_1}\lambda + {w_0}$

• t=$Q{(\left| \lambda \right|)^2}$

Why does $$\det (tI - P{(\lambda )^*}P(\lambda )) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$$ where $p(x, y)$ and $q(x, y)$ are real polynomials in $x, y$?