Aufgaben:Exercise 4.16: Eigenvalues and Eigenvectors - Revision history

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	[[Category:Theory of Stochastic Signals: Exercises\|^4.7 N-dimensionale Zufallsgrößen^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.7 N-dimensionale Zufallsgrößen^]]
	[[de:Aufgaben:4.~~16_Eigenwerte_und_Eigenvektoren~~]]		[[de:Aufgaben:Aufgabe 4.16: Eigenwerte und Eigenvektoren]]

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 The eigenvectors are obtained by substituting the eigenvalues&nbsp; $\lambda_1$&nbsp; and&nbsp; $\lambda_2$&nbsp; into the correlation matrix:
-:$$\left[ \begin{array}{cc}
+:$$\left[ \begin{array}{cc}1- (1+\rho) & \rho \\\rho & 1- (1+\rho)\end{array} \right]\cdot{\boldsymbol{\eta_1}} = \left[ \begin{array}{cc}-\rho & \rho \\\rho & -\rho\end{array} \right]\cdot \left[ \begin{array}{c}\eta_{11} \\\eta_{12}\end{array} \right]=0$$:$$\Rightarrow\hspace{0.3cm}-\rho \cdot \eta_{11} + \rho \cdot\eta_{12} = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\eta_{11}={\rm const} \cdot\eta_{12}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}{\boldsymbol{\eta_1}}={\rm const}\cdot \left[ \begin{array}{c}1 \\1\end{array} \right];$$:$$\left[ \begin{array}{cc}1- (1-\rho) & \rho \\\rho & 1- (1-\rho)\end{array} \right]\cdot{\boldsymbol{\eta_2}} = \left[ \begin{array}{cc}\rho & \rho \\\rho & \rho\end{array} \right]\cdot \left[ \begin{array}{c}\eta_{21} \\\eta_{22}\end{array} \right]=0$$:$$\Rightarrow\hspace{0.3cm}\rho \cdot \eta_{21} + \rho \cdot\eta_{22} = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\eta_{21}=-{\rm const} \cdot\eta_{22}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}{\boldsymbol{\eta_2}}={\rm const}\cdot \left[ \begin{array}{c}-1 \\1\end{array} \right].$$[[File:P_ID676__Sto_A_4_16_d.png|right|frame|Rotate the coordinate system]]Putting this into the&nbsp; "orthonormal form",&nbsp; the following holds::$${\boldsymbol{\eta_1}}= \frac{1}{\sqrt{2}}\cdot \left[\begin{array}{c}1 \\1\end{array} \right],\hspace{0.5cm}{\boldsymbol{\eta_2}}= \frac{1}{\sqrt{2}}\cdot \left[\begin{array}{c}-1 \\1\end{array} \right].$$
 The sketch illustrates the result:

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	[[File:P_ID677__Sto_A_4_16_g.png\|right\|frame\|Best possible decorrelation]]		[[File:P_ID677__Sto_A_4_16_g.png\|right\|frame\|Best possible decorrelation]]
	The same result is obtained using the eigenvector:		The same result is obtained using the eigenvector:
	:$$\left[ \begin{array}{cc}		:$$\left[ \begin{array}{cc}4-5 & 2 \\2 & 1-5\end{array} \right]\cdot \left[ \begin{array}{c}\zeta_{11} \\\zeta_{12}\end{array}\right]=0 \hspace{0.3cm}\Rightarrow\hspace{0.3cm}-\zeta_{11}=2\zeta_{12}=0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\zeta_{12}={\zeta_{11}}/{2}$$:$$\Rightarrow\hspace{0.3cm}\alpha = \arctan({\zeta_{12}}/{\zeta_{11}}) = \arctan(0.5) \hspace{0.15cm}\underline{= 26.56^\circ}.$$
	4-5 & 2 \\
	2 & 1-5
	\end{array} \right]\cdot \left[ \begin{array}{c}
	\zeta_{11} \\
	\zeta_{12}
	\end{array}
	\right]=0 \hspace{0.3cm}
	\Rightarrow\hspace{0.3cm}-\zeta_{11}=
	2\zeta_{12}=0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\zeta_{12}={\zeta_{11}}/{2}$$:$$\Rightarrow\hspace{0.3cm}\alpha = \arctan
	({\zeta_{12}}/{\zeta_{11}}) = \arctan(0.5) \hspace{0.15cm}\underline{= 26.56^\circ}.$$

	The accompanying sketch shows the joint PDF of the random variable  $\mathbf{z}$:		The accompanying sketch shows the joint PDF of the random variable  $\mathbf{z}$:

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	*Some basics on the application of vectors and matrices can be found on the pages   [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Determinant_of_a_matrix\|Determinant of a Matrix]]   and   [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Inverse_of_a_matrix\|Inverse of a Matrix]] .		*Some basics on the application of vectors and matrices can be found on the pages   [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Determinant_of_a_matrix\|Determinant of a Matrix]]   and   [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Inverse_of_a_matrix\|Inverse of a Matrix]] .
	* According to the page  [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables#Contour_lines_for_correlated_random_variables\|"Contour lines for correlated random variables"]]  the angle $\alpha$ between the old and the new system is given by the following equation:		* According to the page  [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables#Contour_lines_for_correlated_random_variables\|"Contour lines for correlated random variables"]]  the angle $\alpha$ between the old and the new system is given by the following equation:
	:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot		:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot\frac{\sigma_1\cdot\sigma_2}{\sigma_1^2 -\sigma_2^2}).$$
	\frac{\sigma_1\cdot\sigma_2}{\sigma_1^2 -\sigma_2^2}).$$
	*In particular,  note:		*In particular,  note:
	**A  $2×2$-covariance matrix has two real eigenvalues  $\lambda_1$  and  $\lambda_2$.		**A  $2×2$-covariance matrix has two real eigenvalues  $\lambda_1$  and  $\lambda_2$.
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	'''(2)'''  In this case, the governing equation is:		'''(2)'''  In this case, the governing equation is:
	:$${\rm det}\left[ \begin{array}{cc}		:$${\rm det}\left[ \begin{array}{cc}1- \lambda & 0 \\0 & 1- \lambda\end{array} \right] = 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}(1- \lambda)^2 = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm} \hspace{0.15cm}\underline{\lambda_{1/2} =1}.$$
	1- \lambda & 0 \\
	0 & 1- \lambda
	\end{array} \right] = 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	(1- \lambda)^2 = 0\hspace{0.3cm}\Rightarrow
	\hspace{0.3cm} \hspace{0.15cm}\underline{\lambda_{1/2} =1}.$$



	'''(3)'''  With positive  $\rho$  the governing equation of the eigenvalues is:		'''(3)'''  With positive  $\rho$  the governing equation of the eigenvalues is:
	:$$(1- \lambda)^2 -\rho^2 = 0\hspace{0.5cm}\Rightarrow		:$$(1- \lambda)^2 -\rho^2 = 0\hspace{0.5cm}\Rightarrow\hspace{0.5cm}\lambda^2 - 2\lambda + 1 - \rho^2 =0\hspace{0.5cm}\Rightarrow\hspace{0.5cm}\lambda_{1/2} =1 \pm \rho.$$
	\hspace{0.5cm}\lambda^2 - 2\lambda + 1 - \rho^2 =
	0\hspace{0.5cm}\Rightarrow\hspace{0.5cm}\lambda_{1/2} =1 \pm \rho.$$

	*For  $\rho= 0.5$  one gets  $\underline{\lambda_{1} =1.5}$  and  $\underline{\lambda_{2} =0.5}$.		*For  $\rho= 0.5$  one gets  $\underline{\lambda_{1} =1.5}$  and  $\underline{\lambda_{2} =0.5}$.
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	\eta_{11} \\		\eta_{11} \\
	\eta_{12}		\eta_{12}
	\end{array} \right]=0$$		\end{array} \right]=0$$:$$\Rightarrow\hspace{0.3cm}-\rho \cdot \eta_{11} + \rho \cdot
	:$$\Rightarrow\hspace{0.3cm}-\rho \cdot \eta_{11} + \rho \cdot
	\eta_{12} = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\eta_{11}=		\eta_{12} = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\eta_{11}=
	{\rm const} \cdot		{\rm const} \cdot
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	1 \\		1 \\
	1		1
	\end{array} \right];$$		\end{array} \right];$$:$$\left[ \begin{array}{cc}
	:$$\left[ \begin{array}{cc}
	1- (1-\rho) & \rho \\		1- (1-\rho) & \rho \\
	\rho & 1- (1-\rho)		\rho & 1- (1-\rho)
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	\eta_{21} \\		\eta_{21} \\
	\eta_{22}		\eta_{22}
	\end{array} \right]=0$$		\end{array} \right]=0$$:$$\Rightarrow\hspace{0.3cm}\rho \cdot \eta_{21} + \rho \cdot
	:$$\Rightarrow\hspace{0.3cm}\rho \cdot \eta_{21} + \rho \cdot
	\eta_{22} = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\eta_{21}=		\eta_{22} = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\eta_{21}=
	-{\rm const} \cdot		-{\rm const} \cdot
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	-1 \\		-1 \\
	1		1
	\end{array} \right].$$		\end{array} \right].$$[[File:P_ID676__Sto_A_4_16_d.png\|right\|frame\|Rotate the coordinate system]]Putting this into the  "orthonormal form",  the following holds::$${\boldsymbol{\eta_1}}= \frac{1}{\sqrt{2}}\cdot \left[

	[[File:P_ID676__Sto_A_4_16_d.png\|right\|frame\|Rotate the coordinate system]]
	Putting this into the  "orthonormal form",  the following holds:
	:$${\boldsymbol{\eta_1}}= \frac{1}{\sqrt{2}}\cdot \left[
	\begin{array}{c}		\begin{array}{c}
	1 \\		1 \\
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	*With  $\sigma_1 = \sigma_2$  almost always results  $($exception:   $\rho= 0)$  the rotation angle  $\alpha = 45^\circ$.		*With  $\sigma_1 = \sigma_2$  almost always results  $($exception:   $\rho= 0)$  the rotation angle  $\alpha = 45^\circ$.
	*This also follows from the equation given in the theory section:		*This also follows from the equation given in the theory section:
	:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot		:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot\frac{\sigma_1\cdot\sigma_2}{\sigma_1^2 -\sigma_2^2})={1}/{2}\cdot \arctan(\infty)\hspace{0.3cm}\rightarrow\hspace{0.3cm}\alpha = 45^\circ.$$
	\frac{\sigma_1\cdot\sigma_2}{\sigma_1^2 -\sigma_2^2})=
	{1}/{2}\cdot \arctan
	(\infty)\hspace{0.3cm}\rightarrow\hspace{0.3cm}\alpha = 45^\circ.$$
	*The eigenvalues  $\lambda_1$  and  $\lambda_2$  do not denote the standard deviations with respect to the new axes, but the variances.		*The eigenvalues  $\lambda_1$  and  $\lambda_2$  do not denote the standard deviations with respect to the new axes, but the variances.

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	'''(6)'''  According to the now familiar scheme:		'''(6)'''  According to the now familiar scheme:
	:$$(4- \lambda) \cdot (1- \lambda) -4 = 0\hspace{0.3cm}\Rightarrow		:$$(4- \lambda) \cdot (1- \lambda) -4 = 0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\lambda^2 - 5\lambda =0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\hspace{0.15cm}\underline{\lambda_{1}=5,\hspace{0.1cm} \lambda_{2} =0}.$$
	\hspace{0.3cm}\lambda^2 - 5\lambda =
	0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\hspace{0.15cm}\underline{\lambda_{1}
	=5,\hspace{0.1cm} \lambda_{2} =0}.$$



	'''(7)'''  According to the equation given on the specification sheet:		'''(7)'''  According to the equation given on the specification sheet:
	:$$\alpha ={1}/{2}\cdot \arctan (2 \cdot 1 \cdot \frac{2 \~~cdot~~		:$$\alpha ={1}/{2}\cdot \arctan (2 \cdot 1 \cdot \frac{2 \cdot1}{2^2 -1^2})= {1}/{2}\cdot \arctan ({4}/{3}) =26.56^\circ.$$
	1}{2^2 -1^2})= {1}/{2}\cdot \arctan ({4}/{3}) =
	26.56^\circ.$$

	[[File:P_ID677__Sto_A_4_16_g.png\|right\|frame\|Best possible decorrelation]]		[[File:P_ID677__Sto_A_4_16_g.png\|right\|frame\|Best possible decorrelation]]
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	\right]=0 \hspace{0.3cm}		\right]=0 \hspace{0.3cm}
	\Rightarrow\hspace{0.3cm}-\zeta_{11}=		\Rightarrow\hspace{0.3cm}-\zeta_{11}=
	2\zeta_{12}=0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\zeta_{12}={\zeta_{11}}/{2}$$		2\zeta_{12}=0\hspace{0.3cm}\Rightarrow\hspace{0.3cm}\zeta_{12}={\zeta_{11}}/{2}$$:$$\Rightarrow\hspace{0.3cm}\alpha = \arctan
	:$$\Rightarrow\hspace{0.3cm}\alpha = \arctan
	({\zeta_{12}}/{\zeta_{11}}) = \arctan(0.5) \hspace{0.15cm}\underline{= 26.56^\circ}.$$		({\zeta_{12}}/{\zeta_{11}}) = \arctan(0.5) \hspace{0.15cm}\underline{= 26.56^\circ}.$$

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	[[Category:Theory of Stochastic Signals: Exercises\|^4.7 N-dimensionale Zufallsgrößen^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.7 N-dimensionale Zufallsgrößen^]]
			[[de:Aufgaben:4.16_Eigenwerte_und_Eigenvektoren]]

Guenter at 13:43, 29 March 2022

2022-03-29T13:43:58Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Correct are <u>proposed solutions 1 and 2</u>:		'''(1)'''  Correct are the  <u>proposed solutions 1 and 2</u>:
	*$\mathbf{K_y}$  is indeed the most general correlation matrix of a 2D random variable with  $\sigma_1 = \sigma_2 = \sigma$.		*$\mathbf{K_y}$  is indeed the most general correlation matrix of a two-dimensional random variable with  $\sigma_1 = \sigma_2 = \sigma$.
	*The parameter  $\rho$  specifies the correlation coefficient.  This can take all values between  $\pm 1$  including these marginal values.		*The parameter  $\rho$  specifies the correlation coefficient.  This can take all values between  $\pm 1$  including these marginal values.

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	'''(2)'''  In this case, the governing equation is:		'''(2)'''  In this case, the governing equation is:
	:$${\rm det}\left[ \begin{array}{cc}		:$${\rm det}\left[ \begin{array}{cc}
	1- \lambda & 0 \		1- \lambda & 0 \\
	0 & 1- \lambda		0 & 1- \lambda
	\end{array} \right] = 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}		\end{array} \right] = 0 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
	(1- \lambda)^2 = 0\hspace{0.3cm}\Rightarrow		(1- \lambda)^2 = 0\hspace{0.3cm}\Rightarrow
	\hspace{0.3cm} \hspace{0.15cm}\underline{\lambda_{1/2} =1}.$$		\hspace{0.3cm} \hspace{0.15cm}\underline{\lambda_{1/2} =1}.$$

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	*For  $\rho= 0.5$  one gets  $\underline{\lambda_{1} =1.5}$  and  $\underline{\lambda_{2} =0.5}$.		*For  $\rho= 0.5$  one gets  $\underline{\lambda_{1} =1.5}$  and  $\underline{\lambda_{2} =0.5}$.
	*By the way, the equation holds in the whole domain of definition  $-1 \le \rho \le +1$.		*By the way,  the equation holds in the whole domain of definition  $-1 \le \rho \le +1$.
	*For  $\rho = 0$  is  $\lambda_1 = \lambda_2 = +1$    ⇒   see subtask  '''(2)''''.		*For  $\rho = 0$   ⇒   $\lambda_1 = \lambda_2 = +1$    ⇒   see subtask  '''(2)'''.
	*For  $\rho = \pm 1$  ~~this gives~~ $\lambda_1 = 2$  and  $\lambda_2 = 0$.		*For  $\rho = \pm 1$   ⇒   $\lambda_1 = 2$  and  $\lambda_2 = 0$.



	'''(4)'''  The eigenvectors are obtained by substituting the eigenvalues  $\lambda_1$  and  $\lambda_2$  into the correlation matrix:		'''(4)'''  Correct are  <u>the proposed solutions 1 and 2</u>.

			The eigenvectors are obtained by substituting the eigenvalues  $\lambda_1$  and  $\lambda_2$  into the correlation matrix:
	:$$\left[ \begin{array}{cc}		:$$\left[ \begin{array}{cc}
	1- (1+\rho) & \rho \\		1- (1+\rho) & \rho \\
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	\end{array} \right].$$		\end{array} \right].$$

	[[File:P_ID676__Sto_A_4_16_d.png\|right\|frame\|~~To rotate~~ the coordinate system]]		[[File:P_ID676__Sto_A_4_16_d.png\|right\|frame\|Rotate the coordinate system]]
	Putting this into ~~what is called~~ orthonormal form, the following holds:		Putting this into the  "orthonormal form",  the following holds:
	:$${\boldsymbol{\eta_1}}= \frac{1}{\sqrt{2}}\cdot \left[		:$${\boldsymbol{\eta_1}}= \frac{1}{\sqrt{2}}\cdot \left[
	\begin{array}{c}		\begin{array}{c}
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	The sketch illustrates the result:		The sketch illustrates the result:
	*The coordinate system defined by  $\mathbf{\eta_1}$  and  $\mathbf{\eta_2}$  is actually in the direction of the ~~principal~~ axes of the original system.		*The coordinate system defined by  $\mathbf{\eta_1}$  and  $\mathbf{\eta_2}$  is actually in the direction of the main axes of the original system.
	*With  $\sigma_1 = \sigma_2$  almost always results  $($exception:   $\rho= 0)$  the angle ~~of rotation~~  $\alpha = 45^\circ$.		*With  $\sigma_1 = \sigma_2$  almost always results  $($exception:   $\rho= 0)$  the rotation angle  $\alpha = 45^\circ$.
	*This also follows from the equation given in the theory section:		*This also follows from the equation given in the theory section:
	:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot		:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot
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	{1}/{2}\cdot \arctan		{1}/{2}\cdot \arctan
	(\infty)\hspace{0.3cm}\rightarrow\hspace{0.3cm}\alpha = 45^\circ.$$		(\infty)\hspace{0.3cm}\rightarrow\hspace{0.3cm}\alpha = 45^\circ.$$
	*The eigenvalues  $\lambda_1$  and  $\lambda_2$  do not denote the ~~scatter~~ with respect to the new axes, but the variances.		*The eigenvalues  $\lambda_1$  and  $\lambda_2$  do not denote the standard deviations with respect to the new axes, but the variances.


	~~Thus, correct are <u>the proposed solutions 1 and 2</u>.~~


	'''(5)'''  By comparing the matrices  $\mathbf{K_x}$  and  $\mathbf{K_z}$  we get.		'''(5)'''  By comparing the matrices   $\mathbf{K_x}$   and   $\mathbf{K_z}$   we get.
	*$\sigma_{1}\hspace{0.15cm}\underline{ =2}$,		*$\sigma_{1}\hspace{0.15cm}\underline{ =2}$,
	*$\sigma_{2}\hspace{0.15cm}\underline{ =1}$,		*$\sigma_{2}\hspace{0.15cm}\underline{ =1}$,
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	The accompanying sketch shows the joint PDF of the random variable  $\mathbf{z}$:		The accompanying sketch shows the joint PDF of the random variable  $\mathbf{z}$:
	*Because  $\rho = 1$  all values lie on the correlation line with coordinates  $z_1$  and  $z_2 = z_1/2$.		*Because of  $\rho = 1$  all values lie on the correlation line with coordinates  $z_1$  and  $z_2 = z_1/2$.
	*By rotating by the angle  $\alpha = \arctan(0.5) = 26.56^\circ$  a new coordinate system is formed.		*By rotating by the angle   $\alpha = \arctan(0.5) = 26.56^\circ$   a new coordinate system is formed.
	*The variance along the axis  $\mathbf{\zeta_1}$ is  $\lambda_1 = 5$  $($~~scatter~~  $\sigma_1 = \sqrt{5} = 2.236)$,		*The variance along the axis   $\mathbf{\zeta_1}$   is  $\lambda_1 = 5$  $($standard deviation  $\sigma_1 = \sqrt{5} = 2.236)$,
	*while in the direction orthogonal to it  $\mathbf{\zeta_2}$  ~~the random variable~~ is not extended  $(\lambda_2 = \sigma_2 = 0)$.		*while in the direction orthogonal to it,  the random variable  $\mathbf{\zeta_2}$  is not extended  $(\lambda_2 = \sigma_2 = 0)$.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 13:22, 29 March 2022

2022-03-29T13:22:54Z

← Older revision		Revision as of 15:22, 29 March 2022
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	[[File:P_ID671__Sto_A_4_16.png\|right\|frame\|Three correlation matrices]]		[[File:P_ID671__Sto_A_4_16.png\|right\|frame\|Three correlation matrices]]
	Although the description of Gaussian random variables using vectors and matrices is actually only necessary and makes sense for more than  $N = 2$  dimensions, here we restrict ourselves to the special case of two-dimensional random variables for simplicity.		Although the description of Gaussian random variables using vectors and matrices is actually only necessary and makes sense for more than  $N = 2$  dimensions,  here we restrict ourselves to the special case of two-dimensional random variables for simplicity.

	In the graph above, the general correlation matrix  $\mathbf{K_x}$  of the 2D–random variable  $\mathbf{x} = (x_1, x_2)^{\rm T}$  is given, where  $\sigma_1^2$  and  $\sigma_2^2$  describe the variances of the individual components.   $\rho$  denotes the correlation coefficient between the two components.

	The random variables  $\mathbf{y}$  and  $\mathbf{z}$  give two special cases of  $\mathbf{x}$  whose process parameters are to be determined from the correlation matrices  $\mathbf{K_y}$  and  $\mathbf{K_z}$  respectively.



			In the graph above,  the general correlation matrix  $\mathbf{K_x}$  of the two-dimensional random variable   $\mathbf{x} = (x_1, x_2)^{\rm T}$   is given,  where  $\sigma_1^2$  and  $\sigma_2^2$  describe the variances of the individual components.   $\rho$  denotes the correlation coefficient between the two components.

			The random variables   $\mathbf{y}$   and   $\mathbf{z}$   give two special cases of  $\mathbf{x}$  whose process parameters are to be determined from the correlation matrices   $\mathbf{K_y}$   and   $\mathbf{K_z}$   respectively.


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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables\|Generalization to N-Dimensional Random Variables]].		*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables\|Generalization to N-Dimensional Random Variables]].
	*Some basics on the application of vectors and matrices can be found on the pages  [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Determinant_of_a_matrix\|Determinant of a Matrix]]  and  [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Inverse_of_a_matrix\|Inverse of a Matrix]] .		*Some basics on the application of vectors and matrices can be found on the pages   [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Determinant_of_a_matrix\|Determinant of a Matrix]]   and   [[Theory_of_Stochastic_Signals/Generalization_to_N-Dimensional_Random_Variables#Basics_of_matrix_operations:_Inverse_of_a_matrix\|Inverse of a Matrix]] .
	* According to the page  [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables#Contour_lines_for_correlated_random_variables\|Contour lines for correlated random variables]]  the angle $\alpha$ between the old and the new system is given by the following equation:		* According to the page  [[Theory_of_Stochastic_Signals/Two-Dimensional_Gaussian_Random_Variables#Contour_lines_for_correlated_random_variables\|"Contour lines for correlated random variables"]]  the angle $\alpha$ between the old and the new system is given by the following equation:
	:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot		:$$\alpha = {1}/{2}\cdot \arctan (2 \cdot\rho \cdot
	\frac{\sigma_1\cdot\sigma_2}{\sigma_1^2 -\sigma_2^2}).$$		\frac{\sigma_1\cdot\sigma_2}{\sigma_1^2 -\sigma_2^2}).$$
	*In particular, note:		*In particular,  note:
	**A  $2×2$-covariance matrix has two real eigenvalues  $\lambda_1$  and  $\lambda_2$.		**A  $2×2$-covariance matrix has two real eigenvalues  $\lambda_1$  and  $\lambda_2$.
	**These two eigenvalues determine two eigenvectors  $\xi_1$  and  $\xi_2$.		**These two eigenvalues determine two eigenvectors  $\xi_1$  and  $\xi_2$.
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	\|type="[]"}		\|type="[]"}
	+ $\mathbf{K_y}$  describes all possible two-dimensional random variables with  $\sigma_1 = \sigma_2 = \sigma$.		+ $\mathbf{K_y}$  describes all possible two-dimensional random variables with  $\sigma_1 = \sigma_2 = \sigma$.
	+ The value range of the parameter  $\rho$  is  $-1 \le \rho \le +1$.		+ The value range of the parameter  $\rho$   is  $-1 \le \rho \le +1$.
	- The value range of the parameter  $\rho$  is  $0 < \rho < 1$.		- The value range of the parameter  $\rho$   is  $0 < \rho < 1$.


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	{Give the eigenvalues of  $\mathbf{K_y}$  under the condition  $\sigma = 1$  and  $0 < \rho < 1$  What values result for  $\rho = 0.5 $, assuming  $\lambda_1 \ge \lambda_2$~~ ~~?		{Give the eigenvalues of   $\mathbf{K_y}$   under the condition   $\sigma = 1$   and   $0 < \rho < 1$   What values result for  $\rho = 0.5 $,  assuming  $\lambda_1 \ge \lambda_2$?
	\|type="{}"}		\|type="{}"}
	$\lambda_1 \ = \ $ { 1.5 3% } $\ (\lambda_1 \ge \lambda_2)$		$\lambda_1 \ = \ $ { 1.5 3% } $\ (\lambda_1 \ge \lambda_2)$
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	{Calculate the corresponding eigenvectors  $\mathbf{\eta_1}$  and  $\mathbf{\eta_2}$.  Which of the following statements are true?		{Calculate the corresponding eigenvectors  $\mathbf{\eta_1}$  and  $\mathbf{\eta_2}$.  Which of the following statements are true?
	\|type="[]"}		\|type="[]"}
	+ $\mathbf{\eta_1}$  and  $\mathbf{\eta_2}$  lie in the direction of the ellipse ~~principal~~ axes.		+ $\mathbf{\eta_1}$  and  $\mathbf{\eta_2}$  lie in the direction of the ellipse main axes.
	+ The new coordinates are rotated by  $45^\circ$~~ ~~.		+ The new coordinates are rotated by  $45^\circ$.
	- The ~~scatterings~~ with respect to the new system are  $\lambda_1$  and  $\lambda_2$.		- The standard deviations with respect to the new system are  $\lambda_1$  and  $\lambda_2$.


	{What are the characteristics of the random variable ~~specified by~~  $\mathbf{~~K_z~~}$  $\mathbf{z}$?		{What are the characteristics of the random variable  $\mathbf{z}$  specified by  $\mathbf{K_z}$?
	\|type="{}"}		\|type="{}"}
	$\sigma_1 = \ $ { 2 3% }		$\sigma_1 = \ $ { 2 3% }

Noah at 11:29, 15 March 2022

2022-03-15T11:29:42Z

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Guenter: Guenter moved page Aufgaben:Aufgabe 4.16: Eigenwerte und Eigenvektoren to Aufgaben:Exercise 4.16: Eigenvalues and Eigenvectors

2022-02-28T09:59:54Z

Guenter moved page Aufgaben:Aufgabe 4.16: Eigenwerte und Eigenvektoren to Aufgaben:Exercise 4.16: Eigenvalues and Eigenvectors

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Javier: Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

2021-03-23T12:43:20Z

Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

← Older revision		Revision as of 14:43, 23 March 2021
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	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^4.7 N-dimensionale Zufallsgrößen^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.7 N-dimensionale Zufallsgrößen^]]