Hi, there: I was using *gsl_eigen_nonsymmv *to get the eigenvalue decomposition of the matrix in the attached file but found that the result is not right. Theory says that there is an eigenvalue with value 1.0 and the corresponding eigenvectors should be real. I've also tested this in the package R.
Could you please check? Thanks, Youna *Eigenvalue decompositon result from R:* eigen(a)$values [1] 1.000000e+00 1.110223e-16 -2.691078e-17 -1.540744e-33 -1.022873e-48 [6] -3.798227e-64 9.041170e-80 -4.313994e-112 1.043900e-127 2.451083e-129 [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [21] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [26] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 [31] 0.000000e+00 eigen(a)$vectors[,1] [1] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 [8] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 [15] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 [22] -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 -0.1796053 [29] -0.1796053 -0.1796053 -0.1796053 *EigenValue decomposition from GSL:* eigenvalue[0] = 1.00000000000000066613 + 0.00000000000000000000 i eigenvalue[1] = -0.00000000000000003534 + 0.00000000000000000000 i eigenvalue[2] = 0.00000000000000000413 + 0.00000000000000000000 i eigenvalue[3] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[4] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[5] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[6] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[7] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[8] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[9] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[10] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[11] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[12] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[13] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[14] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[15] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[16] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[17] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[18] = -0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[19] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[20] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[21] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[22] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[23] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[24] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[25] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[26] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[27] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[28] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[29] = 0.00000000000000000000 + 0.00000000000000000000 i eigenvalue[30] = 0.00000000000000000000 + 0.00000000000000000000 i *The eigenvectors:* The eigenvector [0] = 0,0 The eigenvector [1] = 0.158848,-0.000727072 The eigenvector [2] = 2.0093e-07,-9.19685e-10 The eigenvector [3] = 1.87534e-06,-8.58373e-09 The eigenvector [4] = 1.26586e-05,-5.79402e-08 The eigenvector [5] = 6.58246e-05,-3.01289e-07 The eigenvector [6] = 0.000274269,-1.25537e-06 The eigenvector [7] = 0.000940351,-4.30413e-06 The eigenvector [8] = 0.00270351,-1.23744e-05 The eigenvector [9] = 0.00660858,-3.02484e-05 The eigenvector [10] = 0.013878,-6.35217e-05 The eigenvector [11] = 0.0252328,-0.000115494 The eigenvector [12] = 0.0399519,-0.000182866 The eigenvector [13] = 0.055318,-0.000253199 The eigenvector [14] = 0.0671718,-0.000307455 The eigenvector [15] = 0.0716499,-0.000327952 The eigenvector [16] = 0.0671718,-0.000307455 The eigenvector [17] = 0.055318,-0.000253199 The eigenvector [18] = 0.0399519,-0.000182866 The eigenvector [19] = -0.974767,-0.000115494 The eigenvector [20] = 0.013878,0.025667 The eigenvector [21] = 0.00660858,0.0349227 The eigenvector [22] = 0.00270351,-1.15514e-05 The eigenvector [23] = 0.000940351,-0.625902 The eigenvector [24] = 0.000274269,0.262268 The eigenvector [25] = 6.58244e-05,0.65909 The eigenvector [26] = 1.26585e-05,1.9919e-05 The eigenvector [27] = 1.87525e-06,0.351666 The eigenvector [28] = 2.0093e-07,-9.19685e-10 The eigenvector [29] = 1.38572e-08,-6.34266e-11 The eigenvector [30] = 4.61907e-10,-2.11422e-12
9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 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4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 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2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 0.000132719054818154 2.55228951573373e-05 3.78116965293883e-06 4.05125319957733e-07 2.79396772384642e-08 9.31322574615479e-10 9.31322574615479e-10 2.79396772384642e-08 4.05125319957732e-07 3.78116965293882e-06 2.55228951573373e-05 0.000132719054818153 0.000552996061742303 0.00189598649740219 0.00545096118003132 0.0133245717734099 0.0279816007241608 0.0508756376802922 0.0805530929937958 0.111535051837563 0.13543542008847 0.144464448094368 0.13543542008847 0.111535051837564 0.0805530929937959 0.0508756376802922 0.0279816007241608 0.0133245717734099 0.00545096118003132 0.00189598649740219 0.000552996061742303 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