Eiganvectors from Eiganvalues

This is a unbelievable discovery from PETER B. DENTON, STEPHEN J. PARKE, TERENCE TAO, AND XINING ZHANG. The original paper you can find here. In a short nut, we can get eiganvector through eiganvalues only. This magic formular is like blew:

We only need to know eiganvalues in original matrix then we can calculate its eiganvectors through sub-matrix.

Currently, I don’t know or image what will effect on the road, but the eiganX is the basic for AI, dimension reduction, feature extraction, etc. It also may help us improve the speed to get eiganvector if we need to incrementally add data on a known matrix.

I wrote a very simple python script to prove this formula(surely, it is correct). I think it can archive by GPU as well.

import numpy as np
from numpy import linalg as LA

matrix_size = 6
org_x = np.diag(range(1,matrix_size+1))
w, v = LA.eig(org_x)


print("orgnal matriax: \n %s \n" % org_x)
print("eigan values: \n %s \n" % w)
print("normalized eigenvectors: \n %s \n" % v)

print("START NEW ALGORITHM \n")

result=[]
for _ in range(matrix_size):
    result.append(0)

for n in range(matrix_size):
    for j in range(matrix_size):
        sub_x = np.delete(org_x,j,axis=0)
        sub_x = np.delete(sub_x,j,axis=1)
        w1,v1 = LA.eig(sub_x)

        # in term of new formula to get orignal matrix eigenvecotr through eiganvalue
        numberator = 1
        denominator = 1
        for i in range(matrix_size-1):
            temp_n = w[n] - w1[i]
            numberator = numberator*temp_n

        for i in range(matrix_size):
            if(i!=n):
                temp_d = w[n] - w[i]
                denominator = denominator*temp_d
        result[j] = numberator/denominator

    print("%s \n" % result)

result:

orgnal matriax:
 [[1 0 0 0 0 0]
 [0 2 0 0 0 0]
 [0 0 3 0 0 0]
 [0 0 0 4 0 0]
 [0 0 0 0 5 0]
 [0 0 0 0 0 6]]

eigan values:
 [1. 2. 3. 4. 5. 6.]

normalized eigenvectors:
 [[1. 0. 0. 0. 0. 0.]
 [0. 1. 0. 0. 0. 0.]
 [0. 0. 1. 0. 0. 0.]
 [0. 0. 0. 1. 0. 0.]
 [0. 0. 0. 0. 1. 0.]
 [0. 0. 0. 0. 0. 1.]]

START NEW ALGORITHM

[1.0, -0.0, -0.0, -0.0, -0.0, -0.0]

[0.0, 1.0, -0.0, -0.0, -0.0, -0.0]

[0.0, 0.0, 1.0, -0.0, -0.0, -0.0]

[0.0, 0.0, 0.0, 1.0, -0.0, -0.0]

[0.0, 0.0, 0.0, 0.0, 1.0, -0.0]

[0.0, 0.0, 0.0, 0.0, 0.0, 1.0]

Reference:

EIGENVECTORS FROM EIGENVALUES, https://arxiv.org/pdf/1908.03795.pdf

Tao’s Blog, https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/

Simple way to create Gantt chart for Azure Devops through power BI

usually we use Gantt chart to track the project progress, take the picture blew for example, the dark part means completed, the light part means waiting for being done.

To reach this result, there are two solutions. The steps in azure devops are same for both solutions.

  • Add start date and end date for each tasks. The Admin can change the layout to make these two fields visible on edit page, but I can only change them in backlogs
  • Add Efforts information: estimate time, already done and left hours. The picture blow shows I have done 80% ‘s workload.
  • Create a view for listing all user stories and tasks. ( one time effort for extracting the data)

Power BI steps:

after imported, you will find gantt chart
  • Import data through “Get Data- More – Online Service – Azure Devops(beta)”, then choose the view you created in Devops.
  • set the Task, start date, end date, % complete for Gantt chat.

CRLF vs LF

These days I am coding a Linux shell running for AZcopy, since my laptop is running windows, so I switch between two operations systems. The strange problem came soon:

  1. ./azCopyToAzure.sh: line 7: $’\r’: command not found
  2. source ./config.cfg can not found
  3. when I was using echo ${log_dir}!!!, the result is !!!XXX, basically, the !!! will come to the beginning and overwrite ${log_dir}

After wasting almost two days, I found the problem. It caused by difference between end of sequence in windows and Linux. Here is from wiki:

The term CRLF refers to Carriage Return (ASCII 13, \r) Line Feed (ASCII 10, \n). They’re used to note the termination of a line, however, dealt with differently in today’s popular Operating Systems. For example: in Windows both a CR and LF are required to note the end of a line, whereas in Linux/UNIX a LF is only required. In the HTTP protocol, the CR-LF sequence is always used to terminate a line.

https://www.owasp.org/index.php/CRLF_Injection

In vs code , we have to change this setting at the bottom of right side.

I thought there should not be many person developing bash on windows, but if you are, hope this can solve some of your problems.


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