Spectral Radius, Power Series and Series in general

Spectral Radius and Spectrum of a matrix are formed by the Eigen vectors and values. I started reviewing this concept that I documented as a take away in my earlier post. While reviewing what is Spectral Radius and What is it significance, I came across a Quora Response, which gave me hint that it is related to Power Series.

So what is it (Spectrum & Spectral Radius) Significance?

I could see that any series could help us to approximation of values. And please note that there are 2 kinds of series,
1. Convergence Series - A Series which converge to a finite number (real or complex does not matter)
2. Divergent Series - A series which becomes infinite.

Spectral radius has an impact on Convergence of Matrix Power Sequence M^n and Matrix Series.

if the power series ∑j=0∞cjxj${\displaystyle \underset{}{\overset{}{=0}}}$converges for some radius of convergence |x|<r$|x|<\mathrm{r.}$


There was a related Question in Quora related to Spectral Decomposition. Yes Spectral Decomposition is a familiar term it also is related to Frequency Decomposition (As Spectrum is essentially a decomposition of Electromagnetic radiation.

Spectral Decomposition is related to Data Mining in term of Analysis of Cube or Multidimensional data as like solving rubik cube collecting all similar colors on particular faces.

While doing Matrix decomposition, we prefer to perform spectral decomposition, this is possible with eigen vectors and values.

What is a Series? What are the other series available and why spectum is compared to power series?

If we are familiar with Arithmetic and Geometric Progression, we definitely know what is a sequence. A summation of Sequence is a Series. Both sequence and series will converge or diverge.

There are many associated function to series like Riemann zeta function, gamma function, polygamma function, polylogarithm  and special functions and polynomials like Bernoulli polynomial and constants like Bernoulli number, Euler number, binomial coefficient etc., which can act like tools to provide Summation of series for application of the series in real life.

Greek mathematician Archimedes seems to have produced first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π. There are also many approximation of π. Please find below picture, I love the way how approximation are done by converging the series to π value.

The geometric series is a simplified form of a larger set of series called the power series. A power series is any series of the following form:

Notice how the power series differs from the geometric series:
In a geometric series, every term has the same coefficient.
In a power series, the coefficients may be different — usually according to a rule that’s specified in the sigma notation.
One can think of a power series as a polynomial with an infinite number of terms. For this reason, many useful features of polynomials carry over to power series.
Matrices are actually polynomials of linear order, so i hope there is no surprise when power series comes into the Picture.

Little bit More about Series

The well know series based on the sequences are as below,

The Power Series is very interesting for Polynomial Analysis as it is related to much of the other interesting series and very much relevant to mining and data science especially in terms of dimensional reduction with spectral decomposition.

I created a Youtube playlist with few videos related to series - check this out