# An example of symmetric, real entry matrix with non-real eigenvalues, according to Mathematica. Is this a bug?

The following matrix is symmetric and all entries are real,

In[147]:= mat == 0.5*(mat+ Transpose[mat])

In[153]:= Table[
Select[mat[[i]], Not[Element[#1, Reals]] &], {i, 1, 30}]


Due to limitation of 30000 char, I don't show the results. The first is True, and 2nd all empty lists.

Here's the matrix.

{{4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.758846257005955*^306,
-6.607783593326043*^306,
4.5933074584793086*^306,
3.563944546335317*^306,
-6.265160288720216*^306,
4.35513770635784*^306,
3.524681398036245*^306,
-6.196138474734244*^306,
4.307158167000686*^306,
3.781304214412078*^306,
-6.647263080472186*^306,
4.620751066491604*^306,
3.51759808243552*^306,
-6.183686511176122*^306,
4.2985023603634954*^306,
4.0378293703075546*^306,
-7.098216006051991*^306,
4.934224624944834*^306,
3.356800284504563*^306,
-5.901015452462095*^306,
4.102007565407017*^306,
3.377816973501901*^306,
-5.93796129255433*^306,
4.1276899444466173*^306,
3.2897456023081654*^306,
-5.783138104313788*^306,
4.0200667913504564*^306},
{-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.607783593326043*^306,
1.161601220982305*^307,
-8.074706861625608*^306,
-6.265160288720216*^306,
1.1013704880374341*^307,
-7.656021426550555*^306,
-6.196138474734244*^306,
1.0892369454859652*^307,
-7.57167681884317*^306,
-6.647263080472186*^306,
1.1685414332070263*^307,
-8.12295076690035*^306,
-6.183686511176122*^306,
1.0870479791149493*^307,
-7.556460528212067*^306,
-7.098216006051991*^306,
1.247815740780918*^307,
-8.674014921926125*^306,
-5.901015452462095*^306,
1.0373564233457479*^307,
-7.211036695069822*^306,
-5.93796129255433*^306,
1.0438512384914352*^307,
-7.256184485442771*^306,
-5.783138104313788*^306,
1.0166344263854468*^307,
-7.066990659287911*^306},
{5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.5933074584793086*^306,
-8.074706861625608*^306,
5.613018454478496*^306,
4.35513770635784*^306,
-7.656021426550555*^306,
5.3219751864106184*^306,
4.307158167000686*^306,
-7.57167681884317*^306,
5.2633442233663465*^306,
4.620751066491604*^306,
-8.12295076690035*^306,
5.646554523993312*^306,
4.2985023603634954*^306,
-7.556460528212067*^306,
5.252766833798561*^306,
4.934224624944834*^306,
-8.674014921926125*^306,
6.029618989956918*^306,
4.102007565407017*^306,
-7.211036695069822*^306,
5.012650333810375*^306,
4.1276899444466173*^306,
-7.256184485442771*^306,
5.044034182770469*^306,
4.0200667913504564*^306,
-7.066990659287911*^306,
4.912518766065072*^306},
{3.758846257005955*^306,
-6.607783593326043*^306,
4.5933074584793086*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.887700554570214*^306,
-6.834300256993171*^306,
4.7507673186573794*^306,
3.8453182708785675*^306,
-6.759795225481607*^306,
4.6989761980640905*^306,
3.98516673397331*^306,
-7.005638847914138*^306,
4.8698709207180616*^306,
3.7732495312052806*^306,
-6.633103521952304*^306,
4.610908249329154*^306,
3.856803722207895*^306,
-6.779985829636917*^306,
4.713011411437522*^306,
3.539871810893023*^306,
-6.222842136972033*^306,
4.325720840731321*^306,
3.4701265213078897*^306,
-6.100234892961072*^306,
4.2404921463551987*^306,
3.3852889203129575*^306,
-5.951096442058326*^306,
4.1368206581470294*^306},
{-6.607783593326043*^306,
1.161601220982305*^307,
-8.074706861625608*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.834300256993171*^306,
1.2014212346634932*^307,
-8.351510063872254*^306,
-6.759795225481607*^306,
1.188323781583981*^307,
-8.260464968239932*^306,
-7.005638847914138*^306,
1.2315413367528236*^307,
-8.560885700381923*^306,
-6.633103521952304*^306,
1.1660522838223707*^307,
-8.105647796437825*^306,
-6.779985829636917*^306,
1.191873145770615*^307,
-8.285137872188832*^306,
-6.222842136972033*^306,
1.0939312588244717*^307,
-7.604308675146666*^306,
-6.100234892961072*^306,
1.0723777799108683*^307,
-7.45448270997731*^306,
-5.951096442058326*^306,
1.0461603040784252*^307,
-7.272235628814887*^306},
{4.5933074584793086*^306,
-8.074706861625608*^306,
5.613018454478496*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.7507673186573794*^306,
-8.351510063872254*^306,
5.80543429186982*^306,
4.6989761980640905*^306,
-8.260464968239932*^306,
5.742145579259999*^306,
4.8698709207180616*^306,
-8.560885700381923*^306,
5.950978809062406*^306,
4.610908249329154*^306,
-8.105647796437825*^306,
5.634526608406059*^306,
4.713011411437522*^306,
-8.285137872188832*^306,
5.759296600041806*^306,
4.325720840731321*^306,
-7.604308675146666*^306,
5.2860277974916044*^306,
4.2404921463551987*^306,
-7.45448270997731*^306,
5.181878393449171*^306,
4.1368206581470294*^306,
-7.272235628814887*^306,
5.0551919084324786*^306},
{3.563944546335317*^306,
-6.265160288720216*^306,
4.35513770635784*^306,
3.887700554570214*^306,
-6.834300256993171*^306,
4.7507673186573794*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.7052413677982366*^306,
-6.513550021849522*^306,
4.527802321909851*^306,
3.9276657736932985*^306,
-6.904556261407554*^306,
4.79960478806323*^306,
4.0207837877598504*^306,
-7.06825108783075*^306,
4.9133949351684666*^306,
3.6262148204805167*^306,
-6.374626988796509*^306,
4.4312319371712406*^306,
3.7667170306425864*^306,
-6.621619851940232*^306,
4.602925538277533*^306,
3.6425108735207126*^306,
-6.403274287609083*^306,
4.45114570242136*^306,
3.567133553182516*^306,
-6.270766335278956*^306,
4.359034670461939*^306},
{-6.265160288720216*^306,
1.1013704880374341*^307,
-7.656021426550555*^306,
-6.834300256993171*^306,
1.2014212346634932*^307,
-8.351510063872254*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.513550021849522*^306,
1.145035631304821*^307,
-7.95955350415708*^306,
-6.904556261407554*^306,
1.213771739088534*^307,
-8.437362851407276*^306,
-7.06825108783075*^306,
1.2425481219037526*^307,
-8.637397813125304*^306,
-6.374626988796509*^306,
1.1206139530616148*^307,
-7.789789656357341*^306,
-6.621619851940232*^306,
1.1640335365497068*^307,
-8.091614759832817*^306,
-6.403274287609083*^306,
1.125649943845596*^307,
-7.824796635175863*^306,
-6.270766335278956*^306,
1.1023559910333393*^307,
-7.662872011466831*^306},
{4.35513770635784*^306,
-7.656021426550555*^306,
5.3219751864106184*^306,
4.7507673186573794*^306,
-8.351510063872254*^306,
5.80543429186982*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.527802321909851*^306,
-7.95955350415708*^306,
5.5329712240783717*^306,
4.79960478806323*^306,
-8.437362851407276*^306,
5.865113644824743*^306,
4.9133949351684666*^306,
-8.637397813125304*^306,
6.004165123832632*^306,
4.4312319371712406*^306,
-7.789789656357341*^306,
5.414962282461961*^306,
4.602925538277533*^306,
-8.091614759832817*^306,
5.624771741166156*^306,
4.45114570242136*^306,
-7.824796635175863*^306,
5.4392968894651854*^306,
4.359034670461939*^306,
-7.662872011466831*^306,
5.326737273780226*^306},
{3.524681398036245*^306,
-6.196138474734244*^306,
4.307158167000686*^306,
3.8453182708785675*^306,
-6.759795225481607*^306,
4.6989761980640905*^306,
3.7052413677982366*^306,
-6.513550021849522*^306,
4.527802321909851*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.6799284697010196*^306,
-6.469051752617422*^306,
4.4968699783991354*^306,
3.571390908322127*^306,
-6.278250461928206*^306,
4.3642371554211796*^306,
3.6390574568563934*^306,
-6.39720342745276*^306,
4.446925629708386*^306,
3.346598331547328*^306,
-5.883081146890257*^306,
4.089540786133408*^306,
3.241824187980187*^306,
-5.69889567626151*^306,
3.9615068570505863*^306,
3.170464507445024*^306,
-5.573450448118335*^306,
3.8743053780792307*^306},
{-6.196138474734244*^306,
1.0892369454859652*^307,
-7.57167681884317*^306,
-6.759795225481607*^306,
1.188323781583981*^307,
-8.260464968239932*^306,
-6.513550021849522*^306,
1.145035631304821*^307,
-7.95955350415708*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.469051752617422*^306,
1.1372131529894273*^307,
-7.905176650735032*^306,
-6.278250461928206*^306,
1.1036716471124116*^307,
-7.67201760892087*^306,
-6.39720342745276*^306,
1.1245827299348496*^307,
-7.817378048373208*^306,
-5.883081146890257*^306,
1.0342037003554311*^307,
-7.189120986388672*^306,
-5.69889567626151*^306,
1.001825208452996*^307,
-6.964046471993284*^306,
-5.573450448118335*^306,
9.797728321728929*^306,
-6.810752492228423*^306},
{4.307158167000686*^306,
-7.57167681884317*^306,
5.2633442233663465*^306,
4.6989761980640905*^306,
-8.260464968239932*^306,
5.742145579259999*^306,
4.527802321909851*^306,
-7.95955350415708*^306,
5.5329712240783717*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.4968699783991354*^306,
-7.905176650735032*^306,
5.4951719222603755*^306,
4.3642371554211796*^306,
-7.67201760892087*^306,
5.333094706708313*^306,
4.446925629708386*^306,
-7.817378048373208*^306,
5.434139963604837*^306,
4.089540786133408*^306,
-7.189120986388672*^306,
4.9974159383854617*^306,
3.9615068570505863*^306,
-6.964046471993284*^306,
4.840958567909506*^306,
3.8743053780792307*^306,
-6.810752492228423*^306,
4.734398422491745*^306},
{3.781304214412078*^306,
-6.647263080472186*^306,
4.620751066491604*^306,
3.98516673397331*^306,
-7.005638847914138*^306,
4.8698709207180616*^306,
3.9276657736932985*^306,
-6.904556261407554*^306,
4.79960478806323*^306,
3.6799284697010196*^306,
-6.469051752617422*^306,
4.4968699783991354*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.874115888489161*^306,
-6.810419383045777*^306,
4.7341668663471124*^306,
3.829920695457503*^306,
-6.732727386233149*^306,
4.680160371826918*^306,
3.662471097047468*^306,
-6.438362936763964*^306,
4.4755370814056626*^306,
3.6193421980869895*^306,
-6.362545408867447*^306,
4.42283359748372*^306,
3.527677039492444*^306,
-6.201404598728772*^306,
4.310818838734212*^306},
{-6.647263080472186*^306,
1.1685414332070263*^307,
-8.12295076690035*^306,
-7.005638847914138*^306,
1.2315413367528236*^307,
-8.560885700381923*^306,
-6.904556261407554*^306,
1.213771739088534*^307,
-8.437362851407276*^306,
-6.469051752617422*^306,
1.1372131529894273*^307,
-7.905176650735032*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.810419383045777*^306,
1.1972231473708893*^307,
-8.322327652856341*^306,
-6.732727386233149*^306,
1.183565448524751*^307,
-8.227388087887848*^306,
-6.438362936763964*^306,
1.1318182780722336*^307,
-7.867674939541377*^306,
-6.362545408867447*^306,
1.1184900974905624*^307,
-7.775025974258095*^306,
-6.201404598728772*^306,
1.0901626925198218*^307,
-7.578112018627215*^306},
{4.620751066491604*^306,
-8.12295076690035*^306,
5.646554523993312*^306,
4.8698709207180616*^306,
-8.560885700381923*^306,
5.950978809062406*^306,
4.79960478806323*^306,
-8.437362851407276*^306,
5.865113644824743*^306,
4.4968699783991354*^306,
-7.905176650735032*^306,
5.4951719222603755*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.7341668663471124*^306,
-8.322327652856341*^306,
5.7851485509277*^306,
4.680160371826918*^306,
-8.227388087887848*^306,
5.719152652951408*^306,
4.4755370814056626*^306,
-7.867674939541377*^306,
5.4691031372739043*^306,
4.42283359748372*^306,
-7.775025974258095*^306,
5.4046995173239537*^306,
4.310818838734212*^306,
-7.578112018627215*^306,
5.267817561626806*^306},
{3.51759808243552*^306,
-6.183686511176122*^306,
4.2985023603634954*^306,
3.7732495312052806*^306,
-6.633103521952304*^306,
4.610908249329154*^306,
4.0207837877598504*^306,
-7.06825108783075*^306,
4.9133949351684666*^306,
3.571390908322127*^306,
-6.278250461928206*^306,
4.3642371554211796*^306,
3.874115888489161*^306,
-6.810419383045777*^306,
4.7341668663471124*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.557024609355578*^306,
-6.252995533123645*^306,
4.346681548279548*^306,
3.9061259995270014*^306,
-6.866690874901065*^306,
4.773283199318385*^306,
3.774323999670507*^306,
-6.634992360870052*^306,
4.6122212490311296*^306,
3.6997598931182865*^306,
-6.503913980367406*^306,
4.5211039637410505*^306},
{-6.183686511176122*^306,
1.0870479791149493*^307,
-7.556460528212067*^306,
-6.633103521952304*^306,
1.1660522838223707*^307,
-8.105647796437825*^306,
-7.06825108783075*^306,
1.2425481219037526*^307,
-8.637397813125304*^306,
-6.278250461928206*^306,
1.1036716471124116*^307,
-7.67201760892087*^306,
-6.810419383045777*^306,
1.1972231473708893*^307,
-8.322327652856341*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.252995533123645*^306,
1.0992320107773431*^307,
-7.64115610384464*^306,
-6.866690874901065*^306,
1.2071152742425408*^307,
-8.391091375968692*^306,
-6.634992360870052*^306,
1.1663843282306214*^307,
-8.10795595625484*^306,
-6.503913980367406*^306,
1.1433416839481964*^307,
-7.947778268304451*^306},
{4.2985023603634954*^306,
-7.556460528212067*^306,
5.252766833798561*^306,
4.610908249329154*^306,
-8.105647796437825*^306,
5.634526608406059*^306,
4.9133949351684666*^306,
-8.637397813125304*^306,
6.004165123832632*^306,
4.3642371554211796*^306,
-7.67201760892087*^306,
5.333094706708313*^306,
4.7341668663471124*^306,
-8.322327652856341*^306,
5.7851485509277*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.346681548279548*^306,
-7.64115610384464*^306,
5.31164176724012*^306,
4.773283199318385*^306,
-8.391091375968692*^306,
5.832948682058476*^306,
4.6122212490311296*^306,
-8.10795595625484*^306,
5.636131093110062*^306,
4.5211039637410505*^306,
-7.947778268304451*^306,
5.524785835149796*^306},
{4.0378293703075546*^306,
-7.098216006051991*^306,
4.934224624944834*^306,
3.856803722207895*^306,
-6.779985829636917*^306,
4.713011411437522*^306,
3.6262148204805167*^306,
-6.374626988796509*^306,
4.4312319371712406*^306,
3.6390574568563934*^306,
-6.39720342745276*^306,
4.446925629708386*^306,
3.829920695457503*^306,
-6.732727386233149*^306,
4.680160371826918*^306,
3.557024609355578*^306,
-6.252995533123645*^306,
4.346681548279548*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.3705045411254564*^306,
-5.92510655804788*^306,
4.118754157272097*^306,
3.3643796587281805*^306,
-5.914339451694345*^306,
4.1112695553293383*^306,
3.276171753109707*^306,
-5.759276245674403*^306,
4.0034795572630436*^306},
{-7.098216006051991*^306,
1.247815740780918*^307,
-8.674014921926125*^306,
-6.779985829636917*^306,
1.191873145770615*^307,
-8.285137872188832*^306,
-6.374626988796509*^306,
1.1206139530616148*^307,
-7.789789656357341*^306,
-6.39720342745276*^306,
1.1245827299348496*^307,
-7.817378048373208*^306,
-6.732727386233149*^306,
1.183565448524751*^307,
-8.227388087887848*^306,
-6.252995533123645*^306,
1.0992320107773431*^307,
-7.64115610384464*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-5.92510655804788*^306,
1.041591467860754*^307,
-7.24047600899865*^306,
-5.914339451694345*^306,
1.0396986873678598*^307,
-7.227318612675841*^306,
-5.759276245674403*^306,
1.012439681848351*^307,
-7.037831484964891*^306},
{4.934224624944834*^306,
-8.674014921926125*^306,
6.029618989956918*^306,
4.713011411437522*^306,
-8.285137872188832*^306,
5.759296600041806*^306,
4.4312319371712406*^306,
-7.789789656357341*^306,
5.414962282461961*^306,
4.446925629708386*^306,
-7.817378048373208*^306,
5.434139963604837*^306,
4.680160371826918*^306,
-8.227388087887848*^306,
5.719152652951408*^306,
4.346681548279548*^306,
-7.64115610384464*^306,
5.31164176724012*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.118754157272097*^306,
-7.24047600899865*^306,
5.0331146571847644*^306,
4.1112695553293383*^306,
-7.227318612675841*^306,
5.023968478922393*^306,
4.0034795572630436*^306,
-7.037831484964891*^306,
4.892249177781899*^306},
{3.356800284504563*^306,
-5.901015452462095*^306,
4.102007565407017*^306,
3.539871810893023*^306,
-6.222842136972033*^306,
4.325720840731321*^306,
3.7667170306425864*^306,
-6.621619851940232*^306,
4.602925538277533*^306,
3.346598331547328*^306,
-5.883081146890257*^306,
4.089540786133408*^306,
3.662471097047468*^306,
-6.438362936763964*^306,
4.4755370814056626*^306,
3.9061259995270014*^306,
-6.866690874901065*^306,
4.773283199318385*^306,
3.3705045411254564*^306,
-5.92510655804788*^306,
4.118754157272097*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
3.9560394849721805*^306,
-6.954435221878715*^306,
4.8342774431097386*^306,
3.920003545893777*^306,
-6.891086611498986*^306,
4.7902415511299526*^306},
{-5.901015452462095*^306,
1.0373564233457479*^307,
-7.211036695069822*^306,
-6.222842136972033*^306,
1.0939312588244717*^307,
-7.604308675146666*^306,
-6.621619851940232*^306,
1.1640335365497068*^307,
-8.091614759832817*^306,
-5.883081146890257*^306,
1.0342037003554311*^307,
-7.189120986388672*^306,
-6.438362936763964*^306,
1.1318182780722336*^307,
-7.867674939541377*^306,
-6.866690874901065*^306,
1.2071152742425408*^307,
-8.391091375968692*^306,
-5.92510655804788*^306,
1.041591467860754*^307,
-7.24047600899865*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-6.954435221878715*^306,
1.2225401045421406*^307,
-8.498314905704879*^306,
-6.891086611498986*^306,
1.2114038707164825*^307,
-8.42090294877816*^306},
{4.102007565407017*^306,
-7.211036695069822*^306,
5.012650333810375*^306,
4.325720840731321*^306,
-7.604308675146666*^306,
5.2860277974916044*^306,
4.602925538277533*^306,
-8.091614759832817*^306,
5.624771741166156*^306,
4.089540786133408*^306,
-7.189120986388672*^306,
4.9974159383854617*^306,
4.4755370814056626*^306,
-7.867674939541377*^306,
5.4691031372739043*^306,
4.773283199318385*^306,
-8.391091375968692*^306,
5.832948682058476*^306,
4.118754157272097*^306,
-7.24047600899865*^306,
5.0331146571847644*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.8342774431097386*^306,
-8.498314905704879*^306,
5.907483604685052*^306,
4.7902415511299526*^306,
-8.42090294877816*^306,
5.85367177593719*^306},
{3.377816973501901*^306,
-5.93796129255433*^306,
4.1276899444466173*^306,
3.4701265213078897*^306,
-6.100234892961072*^306,
4.2404921463551987*^306,
3.6425108735207126*^306,
-6.403274287609083*^306,
4.45114570242136*^306,
3.241824187980187*^306,
-5.69889567626151*^306,
3.9615068570505863*^306,
3.6193421980869895*^306,
-6.362545408867447*^306,
4.42283359748372*^306,
3.774323999670507*^306,
-6.634992360870052*^306,
4.6122212490311296*^306,
3.3643796587281805*^306,
-5.914339451694345*^306,
4.1112695553293383*^306,
3.9560394849721805*^306,
-6.954435221878715*^306,
4.8342774431097386*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306,
4.062093416665989*^306,
-7.140870468744135*^306,
4.963875272375107*^306},
{-5.93796129255433*^306,
1.0438512384914352*^307,
-7.256184485442771*^306,
-6.100234892961072*^306,
1.0723777799108683*^307,
-7.45448270997731*^306,
-6.403274287609083*^306,
1.125649943845596*^307,
-7.824796635175863*^306,
-5.69889567626151*^306,
1.001825208452996*^307,
-6.964046471993284*^306,
-6.362545408867447*^306,
1.1184900974905624*^307,
-7.775025974258095*^306,
-6.634992360870052*^306,
1.1663843282306214*^307,
-8.10795595625484*^306,
-5.914339451694345*^306,
1.0396986873678598*^307,
-7.227318612675841*^306,
-6.954435221878715*^306,
1.2225401045421406*^307,
-8.498314905704879*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306,
-7.140870468744135*^306,
1.2553140910588514*^307,
-8.726138645064861*^306},
{4.1276899444466173*^306,
-7.256184485442771*^306,
5.044034182770469*^306,
4.2404921463551987*^306,
-7.45448270997731*^306,
5.181878393449171*^306,
4.45114570242136*^306,
-7.824796635175863*^306,
5.4392968894651854*^306,
3.9615068570505863*^306,
-6.964046471993284*^306,
4.840958567909506*^306,
4.42283359748372*^306,
-7.775025974258095*^306,
5.4046995173239537*^306,
4.6122212490311296*^306,
-8.10795595625484*^306,
5.636131093110062*^306,
4.1112695553293383*^306,
-7.227318612675841*^306,
5.023968478922393*^306,
4.8342774431097386*^306,
-8.498314905704879*^306,
5.907483604685052*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306,
4.963875272375107*^306,
-8.726138645064861*^306,
6.06585206007421*^306},
{3.2897456023081654*^306,
-5.783138104313788*^306,
4.0200667913504564*^306,
3.3852889203129575*^306,
-5.951096442058326*^306,
4.1368206581470294*^306,
3.567133553182516*^306,
-6.270766335278956*^306,
4.359034670461939*^306,
3.170464507445024*^306,
-5.573450448118335*^306,
3.8743053780792307*^306,
3.527677039492444*^306,
-6.201404598728772*^306,
4.310818838734212*^306,
3.6997598931182865*^306,
-6.503913980367406*^306,
4.5211039637410505*^306,
3.276171753109707*^306,
-5.759276245674403*^306,
4.0034795572630436*^306,
3.920003545893777*^306,
-6.891086611498986*^306,
4.7902415511299526*^306,
4.062093416665989*^306,
-7.140870468744135*^306,
4.963875272375107*^306,
4.1715201480433983*^306,
-7.333234856864358*^306,
5.097594660460223*^306},
{-5.783138104313788*^306,
1.0166344263854468*^307,
-7.066990659287911*^306,
-5.951096442058326*^306,
1.0461603040784252*^307,
-7.272235628814887*^306,
-6.270766335278956*^306,
1.1023559910333393*^307,
-7.662872011466831*^306,
-5.573450448118335*^306,
9.797728321728929*^306,
-6.810752492228423*^306,
-6.201404598728772*^306,
1.0901626925198218*^307,
-7.578112018627215*^306,
-6.503913980367406*^306,
1.1433416839481964*^307,
-7.947778268304451*^306,
-5.759276245674403*^306,
1.012439681848351*^307,
-7.037831484964891*^306,
-6.891086611498986*^306,
1.2114038707164825*^307,
-8.42090294877816*^306,
-7.140870468744135*^306,
1.2553140910588514*^307,
-8.726138645064861*^306,
-7.333234856864358*^306,
1.2891303783143318*^307,
-8.961207790830412*^306},
{4.0200667913504564*^306,
-7.066990659287911*^306,
4.912518766065072*^306,
4.1368206581470294*^306,
-7.272235628814887*^306,
5.0551919084324786*^306,
4.359034670461939*^306,
-7.662872011466831*^306,
5.326737273780226*^306,
3.8743053780792307*^306,
-6.810752492228423*^306,
4.734398422491745*^306,
4.310818838734212*^306,
-7.578112018627215*^306,
5.267817561626806*^306,
4.5211039637410505*^306,
-7.947778268304451*^306,
5.524785835149796*^306,
4.0034795572630436*^306,
-7.037831484964891*^306,
4.892249177781899*^306,
4.7902415511299526*^306,
-8.42090294877816*^306,
5.85367177593719*^306,
4.963875272375107*^306,
-8.726138645064861*^306,
6.06585206007421*^306,
5.097594660460223*^306,
-8.961207790830412*^306,
6.229257057416106*^306}}


If you we do Eigenvalues[mat], then you'll get non-real eigenvalues. How can that be?

Out[148]= {(...), -1.58298*10^291 +
6.14057*10^290 I, -1.58298*10^291 - 6.14057*10^290 I,
(...), 8.28201*10^289 + 3.31445*10^290 I,
8.28201*10^289 - 3.31445*10^290 I}

• Also, Mathematica doesn't show any error message when computing this... i.e. it won't say that it's ill-conditioned. – An old man in the sea. Apr 17 at 19:52
• With large exponents you just have to standardize a bit: 10^306 Eigenvalues[m/10^306] works fine. – JimB Apr 17 at 20:15
• @JimB So, it's a bug? By the way, thanks for the comment. How would I make that automatic. This matrix showed during a simulation. – An old man in the sea. Apr 17 at 20:31
• @JimB I've tried dividing by the 'Max[mat]' , but it gives different values by something of the order of 10^-15 . – An old man in the sea. Apr 17 at 21:09
• How about Eigenvalues[mat/Norm[mat]]*Norm[mat]? Does should automatically adapt to the range of number and does not produce imaginary numbers, at least in my computer. – Mauricio Fernández Apr 18 at 7:16

Be aware that $MaxMachineNumber is around 10^308, in your situation, fed Mathematica with machine-precision-numbers, the underline hardware arithmetic will be called, and the intermediate results are very likely to be overflowed. Luckily, you can renormalize your matrix, as suggested by JimB in the comment. newMat = mat/10^305; 10^305 Eigenvalues@newMat  BTW, my experience tells me, even if the matrix itself is not ill-conditioned, the problem is practically ill-posted. Since one would expect the absolute error of each entity is around 10^305*$MachineEpsilon~10^289, which is enormous, you may probably go back and check whether the entities are trustable or check whether the large rescaling constant can be absorbed into a meaningful quantity.