A few questions in the article, "Dynamic identification and damage assessment of the bell tower"

1. Why the "rank" of the matrix matters in SVD?

 - The number of the diagonal non-zero elements of the singular value matrix obtained SVD determines the rank;

 - In SSI, choosing the dimension (or rank) of the system matrix has to be done by the user;

 - the rank means the number of linearly independent column vectors in the matrix;

 - the size of the system matrix is equal to the size of the eigenvalue (or singular value) matrix. In other words, the user chooses the number of the modes which will be taken into account in the system identification;

2. What on earth is "Rank" in matrix and why does it matter?

 - answered above.

3. Why the "complex conjugate" matters in dynamic identification?

 - The complex number is the product of FFT/DFT;

 - In FDD method, the cross-spectrum is calculated by applying DFT to the cross-correlation of the signals. Thus, the cross-spectrum contains complex terms;

 - The expectation (average) of the PSD/Cross Spectrum density at each frequency interval (df) multiplied by its complex conjugate becomes the amplitude of impulse response. The higher amplitudes (peaks) at corresponding frequency range indicates dominant frequency contents.  The series of the impulse responses form PSD in frequency-domain.  

source: Barry Van Veen, Coherence and the Cross Spectrum ( https://youtu.be/igRrGrxbg-Y )

4. Why correlation matters in the spectral analysis?

 - the correlation is calculated in time-domain. Maybe it might be a key to shift to frequency-domain. 

5. Why is it called "POWER" spectral density?

 - Power means square (제곱).

 - The correlation spectra are multiplied by their own complex conjugates. Thus, it's almost same as square.

6. What does "Coherence" of the signals mean?

 - we use Coherence to assess the 'quality' of the modal parameter?

7.