Lysario – by Panagiotis Chatzichrisafis
"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"
Archive for the ‘Solved Problems’ Category
Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.2
Author: Panagiotis21 Sep
In exercise [1, p. 34 exercise 2.2] we are asked to
prove that the rows and columns of a unitary matrix are orthonormal as per [1, p. 21,(2.21)].
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Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.1
Author: Panagiotis21 Sep
In exercise [1, p. 34 exercise 2.1] we are asked to prove that the inverse of a lower triangular matrix is also lower triangular by using relation [1, p. 23, (2.30)].
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Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.5
Author: Panagiotis13 Jun
In exercise [1, p. 34 exercise 2.5] we are asked to show that
the matrix 
is circulant
when ![\mathbf{e_i}^H=\left[ {\begin{array}{*{20}c}
1 & {e^{-j2\pi f_i } } & {e^{-j2\pi (2f_i) } } & \cdots & {e^{-j2\pi (n - 1)f_i } }
\end{array}} \right], i=1,2](https://lysario.de/wp-content/cache/tex_38ce66bc9234a41f314618cd54d2cfd9.png)
.
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is circulant
when .
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Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 34 exercise 2.4
Author: Panagiotis1 Feb
The exercise [1, p. 34 exercise 2.4] asks to show that if 
is a full rank 
matrix with 
> 
, 
is a 
vector, and 
is an vector, that the effect of the linear transformation
is to project onto the subspace spanned by the columns of . Specifically, if 
are the columns of , the exercise [1, p. 34 exercise 2.4] asks to show that
Furthermore it is asked why the transform
must be idempotent.
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is a full rank matrix with > , is a vector, and is an vector, that the effect of the linear transformation
 | (1) | ||
is to project
onto the subspace spanned by the columns of . Specifically, if are the columns of , the exercise [1, p. 34 exercise 2.4] asks to show that
 | (2) | ||
Furthermore it is asked why the transform
must be idempotent.
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Liboff: “Introductory Quantum Mechanics”, 2nd edition p.86 exercise 3.16
Author: Panagiotis2 Jan
The exercise [1, p. 86 ex. 3.16] asks to prove that if the eigenfunctions and eigenvalues of an operator 
are 
and 
, respectively (
) then the eigenfunctions of a function 
having an expansion of the form:
will also be
with corresponding eigenvalues 
, 
. That is 
.
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are and , respectively () then the eigenfunctions of a function having an expansion of the form:
 | (1) | ||
will also be
with corresponding eigenvalues , . That is .
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