# Trying to extract the Eigen values and Eigen vector of a matrix

I have a Matrix A, which is a function of ω. I wanted to find the eigenvalues and eigenvectors of this matrix, how to do it. I have used the EigenSystem function before, but the matrix was having just numerical values. Now the matrix is a function of ω. The one method that I have tried is by taking the determinant of the matrix and find the roots. but this is tedious. I want to use Mathematica inbuilt algorithm to extract eigenvalues and eigenvectors.

R={{{-422783.+52.3333 ω^2,-323879.+40.0907 ω^2,-248196.+30.7225 ω^2,-17938.6+2.21696 ω^2,0.,-105373.+13.0433 ω^2,-128795.+15.9427 ω^2,-238817.+29.5579 ω^2,0.,-162874.+20.1611 ω^2,-137384.+17.0021 ω^2,0.,-97613.2+12.0829 ω^2,55692.3 -6.89377 ω^2,-102217.+13.1054 ω^2,0.,0.,0.,0.,0.,0.,0.533402,0.408248},{-323879.+40.0907 ω^2,-2.02144*10^6+43.7642 ω^2,2.22587*10^6+5.09888 ω^2,518519. -11.2358 ω^2,-3.36222*10^6+26.0116 ω^2,0.,306368. +17.7384 ω^2,-1.2836*10^6+27.8179 ω^2,1.64393*10^6-2.51223 ω^2,-1.8965*10^6+24.8127 ω^2,-293086.+6.353 ω^2,0.,0.,1.99167*10^6-8.41918 ω^2,-327225.+7.01249 ω^2,0.,0.,0.,0.,0.,0.,0.620112,-2.24484*10^-12},{-248196.+30.7225 ω^2,2.22587*10^6+5.09888 ω^2,-3.62888*10^6+44.1887 ω^2,-1.73801*10^6+21.1729 ω^2,4.77441*10^6-36.937 ω^2,-1.42148*10^6+23.4841 ω^2,0.,-912657.+11.0855 ω^2,-1.07748*10^6+1.64659 ω^2,-0.000128376,-1.56914*10^6+19.1029 ω^2,2.56405*10^6-1.23979 ω^2,-2.74029*10^6+20.1672 ω^2,0.000170136,-939377.+11.56 ω^2,0.,0.,0.,0.,0.,0.,-2.55864*10^-12,0.657676},{-17938.6+2.21696 ω^2,518519. -11.2358 ω^2,-1.73801*10^6+21.1729 ω^2,-2.54498*10^7+52.2008 ω^2,3.8954*10^6-30.1442 ω^2,-2.54893*10^7+50.3311 ω^2,2.63079*10^7-45.9074 ω^2,6109.24 -0.00489818 ω^2,2.78298*10^7-42.5277 ω^2,-4.30574*10^6+3.46461 ω^2,-793.55+0.00456839 ω^2,-6.04438*10^6+2.92501 ω^2,2.34361*10^6-0.634772 ω^2,6.2272*10^6-1.31025 ω^2,-30830.8+0.00426605 ω^2,0.,0.,0.,0.,0.,0.,4.7964*10^-12,-7.8324*10^-12},{0.,-3.36222*10^6+26.0116 ω^2,4.77441*10^6-36.937 ω^2,3.8954*10^6-30.1442 ω^2,-6.76452*10^6+52.3333 ω^2,3.15034*10^6-24.3724 ω^2,-2.00787*10^6+15.5338 ω^2,-1.24205*10^6+9.62951 ω^2,0.,-2.21169*10^6+17.1106 ω^2,1.64506*10^6-12.723 ω^2,0.,2.28413*10^6-17.671 ω^2,699845. -5.41432 ω^2,705602. -5.54426 ω^2,0.,0.,0.,0.,0.,0.,0.408248,-0.57735},{-105373.+13.0433 ω^2,0.,-1.42148*10^6+23.4841 ω^2,-2.54893*10^7+50.3311 ω^2,3.15034*10^6-24.3724 ω^2,-2.61081*10^7+51.5526 ω^2,2.59334*10^7-41.751 ω^2,-4.64693*10^6+9.18168 ω^2,2.8753*10^7-43.94 ω^2,-9.20734*10^6+11.8767 ω^2,-714864.+1.41353 ω^2,0.0000878348,-0.000390834,8.50009*10^6-3.31961 ω^2,-757990.+1.43654 ω^2,0.,0.,0.,0.,0.,0.,0.118904,-9.55289*10^-12},{-128795.+15.9427 ω^2,306368. +17.7384 ω^2,0.,2.63079*10^7-45.9074 ω^2,-2.00787*10^6+15.5338 ω^2,2.59334*10^7-41.751 ω^2,-2.89568*10^7+50.5308 ω^2,-5.4127*10^6+9.43531 ω^2,-3.00227*10^7+45.8803 ω^2,-0.000401084,-5.87669*10^6+10.2539 ω^2,1.34861*10^7-6.52092 ω^2,-1.11138*10^7+11.9033 ω^2,0.000618427,-3.20802*10^6+5.66126 ω^2,0.,0.,0.,0.,0.,0.,-1.01315*10^-11,0.300783},{-238817.+29.5579 ω^2,-1.2836*10^6+27.8179 ω^2,-912657.+11.0855 ω^2,6109.24 -0.00489818 ω^2,-1.24205*10^6+9.62951 ω^2,-4.64693*10^6+9.18168 ω^2,-5.4127*10^6+9.43531 ω^2,-4.37356*10^7+38.2294 ω^2,4.76406*10^6-7.29091 ω^2,-4.54155*10^7+36.5616 ω^2,20058.4 -0.0163177 ω^2,6.36494*10^7-30.781 ω^2,-1.81545*10^7+4.92544 ω^2,1.00745*10^7-2.1077 ω^2,-52387.8+0.00467981 ω^2,0.,0.,0.,0.,0.,0.,-2.95423*10^-11,-1.07327*10^-11},{0.,1.64393*10^6-2.51223 ω^2,-1.07748*10^6+1.64659 ω^2,2.78298*10^7-42.5277 ω^2,0.,2.8753*10^7-43.94 ω^2,-3.00227*10^7+45.8803 ω^2,4.76406*10^6-7.29091 ω^2,-3.42454*10^7+52.3333 ω^2,1.1472*10^7-17.5314 ω^2,-7.56918*10^6+11.5654 ω^2,0.,-9.59599*10^6+14.6645 ω^2,-2.16264*10^6+3.30492 ω^2,-3.25147*10^6+5.0527 ω^2,0.,0.,0.,0.,0.,0.,-0.220942,0.408248},{-162874.+20.1611 ω^2,-1.8965*10^6+24.8127 ω^2,-0.000128376,-4.30574*10^6+3.46461 ω^2,-2.21169*10^6+17.1106 ω^2,-9.20734*10^6+11.8767 ω^2,-0.000401084,-4.54155*10^7+36.5616 ω^2,1.1472*10^7-17.5314 ω^2,-4.9468*10^7+39.8205 ω^2,1.11342*10^7-8.96176 ω^2,6.7446*10^7-32.612 ω^2,-6.79755*10^6-4.07388 ω^2,0.00179429,5.10607*10^6-4.17185 ω^2,0.,0.,0.,0.,0.,0.,-2.7989*10^-11,-0.20051},{-137384.+17.0021 ω^2,-293086.+6.353 ω^2,-1.56914*10^6+19.1029 ω^2,-793.55+0.00456839 ω^2,1.64506*10^6-12.723 ω^2,-714864.+1.41353 ω^2,-5.87669*10^6+10.2539 ω^2,20058.4 -0.0163177 ω^2,-7.56918*10^6+11.5654 ω^2,1.11342*10^7-8.96176 ω^2,-1.11086*10^8+32.7076 ω^2,-3.08645*10^7+14.9235 ω^2,-1.13489*10^8+30.8457 ω^2,1.43849*10^8-30.2158 ω^2,436654. -0.0597055 ω^2,0.,0.,0.,0.,0.,0.,-2.05928*10^-11,-5.77511*10^-11},{0.,0.,2.56405*10^6-1.23979 ω^2,-6.04438*10^6+2.92501 ω^2,0.,0.0000878348,1.34861*10^7-6.52092 ω^2,6.36494*10^7-30.781 ω^2,0.,6.7446*10^7-32.612 ω^2,-3.08645*10^7+14.9235 ω^2,-1.08232*10^8+52.3333 ω^2,-0.00361532,2.72389*10^7-13.1708 ω^2,-1.84196*10^7+8.96065 ω^2,0.,0.,0.,0.,0.,0.,0.57735,0.},{-97613.2+12.0829 ω^2,0.,-2.74029*10^6+20.1672 ω^2,2.34361*10^6-0.634772 ω^2,2.28413*10^6-17.671 ω^2,-0.000390834,-1.11138*10^7+11.9033 ω^2,-1.81545*10^7+4.92544 ω^2,-9.59599*10^6+14.6645 ω^2,-6.79755*10^6-4.07388 ω^2,-1.13489*10^8+30.8457 ω^2,-0.00361532,-1.27731*10^8+34.7167 ω^2,1.51728*10^8-28.5914 ω^2,2.12322*10^7-5.70772 ω^2,0.,0.,0.,0.,0.,0.,-0.254205,-6.51979*10^-11},{55692.3 -6.89377 ω^2,1.99167*10^6-8.41918 ω^2,0.000170136,6.2272*10^6-1.31025 ω^2,699845. -5.41432 ω^2,8.50009*10^6-3.31961 ω^2,0.000618427,1.00745*10^7-2.1077 ω^2,-2.16264*10^6+3.30492 ω^2,0.00179429,1.43849*10^8-30.2158 ω^2,2.72389*10^7-13.1708 ω^2,1.51728*10^8-28.5914 ω^2,-2.27325*10^8+47.7508 ω^2,-9.90303*10^7+20.7418 ω^2,0.,0.,0.,0.,0.,0.,3.68701*10^-11,0.428263},{-102217.+13.1054 ω^2,-327225.+7.01249 ω^2,-939377.+11.56 ω^2,-30830.8+0.00426605 ω^2,705602. -5.54426 ω^2,-757990.+1.43654 ω^2,-3.20802*10^6+5.66126 ω^2,-52387.8+0.00467981 ω^2,-3.25147*10^6+5.0527 ω^2,5.10607*10^6-4.17185 ω^2,436654. -0.0597055 ω^2,-1.84196*10^7+8.96065 ω^2,2.12322*10^7-5.70772 ω^2,-9.90303*10^7+20.7418 ω^2,-3.574*10^8+47.9839 ω^2,0.,0.,0.,0.,0.,0.,-5.00794*10^-11,-7.77121*10^-11},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,-6.1685*10^11+0.157 ω^2,0.,0.,0.,0.,0.,-1.,0.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,-6.1685*10^11+0.157 ω^2,0.,0.,0.,-1.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.},{0.533402,0.620112,-2.55864*10^-12,4.7964*10^-12,0.408248,0.118904,-1.01315*10^-11,-2.95423*10^-11,-0.220942,-2.7989*10^-11,-2.05928*10^-11,0.57735,-0.254205,3.68701*10^-11,-5.00794*10^-11,-1.,0.,0.,0.,0.,0.,0.,0.},{0.408248,-2.24484*10^-12,0.657676,-7.8324*10^-12,-0.57735,-9.55289*10^-12,0.300783,-1.07327*10^-11,0.408248,-0.20051,-5.77511*10^-11,0.,-6.51979*10^-11,0.428263,-7.77121*10^-11,0.,0.,0.,-1.,0.,0.,0.,0.}}}

• You are asking Mathematica to solve a polynomial equation of degree 26 with variable coefficients. There is no generic formula for this. I fear the best you can do is define a function of ω that does the numerical computation for each required value of ω. Why do you need a symbolic solution? – user64074 Oct 14 '19 at 16:52
• @Jean-Claude Arbaut I don't need a symbolic solution, I need to find the roots of ω. And I used NSolve and it is not the best way to get the roots. – acoustics Oct 14 '19 at 17:08
• The roots of ω? If you computes eigenvalues, you are getting roots of a polynomial whose coefficients depends on ω. That is, ω must be known. – user64074 Oct 14 '19 at 17:10
• @Jean-Claude Arbaut If I have a matrix with numbers, I know that I can use Eigensystem to get the eigenvalue and eigenvector. I wanted to find values of ω such that Det[R] goes to zero using some Mathematica algorithms. – acoustics Oct 14 '19 at 17:28
• Ah, but then these are not eigenvalues, and there is no point in using the Eigensystem function. However, you could indeed computes the roots of the determinant as a polynomial function of ω, and for thoses values, find the kernel of R(ω). – user64074 Oct 14 '19 at 18:14