# Lysario – by Panagiotis Chatzichrisafis

"ούτω γάρ ειδέναι το σύνθετον υπολαμβάνομεν, όταν ειδώμεν εκ τίνων και πόσων εστίν …"

## Archive for the ‘Kay: Modern Spectral Estimation, Theory and Application’ Category

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)]. read the conclusion >
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)]. read the conclusion >
In exercise [1, p. 34 exercise 2.5] we are asked to show that the matrix $\mathbf{R}=\sigma^2 \mathbf{I}+P_1\mathbf{e_1}\mathbf{e_1}^H+P_2\mathbf{e_2}\mathbf{e_2}^H$ is circulant when $\mathbf{e_i}^H=\left[ {\begin{array}{*{20}c} 1 & {e^{-j2\pi f_i } } & {e^{-j2\pi (2f_i) } } & \cdots & {e^{-j2\pi (n - 1)f_i } } \end{array}} \right], i=1,2$. read the conclusion >
The exercise [1, p. 34 exercise 2.4] asks to show that if $\mathbf{F}$ is a full rank $m \times n$ matrix with $m$ > $n$, $\mathbf{x}$ is a $m \times 1$ vector, and $\mathbf{y}$ is an $m \times 1$ vector, that the effect of the linear transformation
 $\mathbf{y}=\mathbf{F}\left(\mathbf{F}^H\mathbf{F}\right)^{-1}\mathbf{F}^H\mathbf{x}$ (1)

is to project $\mathbf{x}$ onto the subspace spanned by the columns of $\mathbf{F}$. Specifically, if $\{\mathbf{f_1},\mathbf{f_2},...,\mathbf{f_n}\}$ are the columns of $\mathbf{F}$, the exercise [1, p. 34 exercise 2.4] asks to show that
 $(\mathbf{x-y})^{H}\mathbf{f_i}=0, \textrm{ i=1,2,...,n.}$ (2)

Furthermore it is asked why the transform $\mathbf{F}\left(\mathbf{F}^H\mathbf{F}\right)^{-1}\mathbf{F}^H$ must be idempotent. read the conclusion >
Exercise [1, p. 14 exercise 1.1] asks to prove that
 $g(x)=\frac{1}{1-x}$ (1)

is a monotonically increasing function over the interval $0. Furthermore it asks to show that due to the monotonicity of the peaks of the spectral estimate shown in  [1, Figure 1.2b] that they must also be present in the spectral estimate of [1, Figure 1.2a]. read the conclusion >