Eigenvalues of a Matrix and RegionPlot

So, this is my problem.

I have a 15 x 15 matrix with 7 parameters. I'm assigning numerical values to 5 of the parameters. Then, I do something like:

RegionPlot[{Max[Re[Eigenvalues[M/.{a->a1, b->b1}]]]>1},{a1,0,1},{b1,0,1}] 

Where M is the matrix and a and b the two parameters. This was working well when one of the five parameters was 0. But, as soon as I give it another values, say 0.5, it does not work anymore. My guess is that Mathematica is trying to get the Eigenvalues first, by solving this:

Max[Re[Eigenvalues[M/.{a->a1, b->b1}]]] 

And only then uses RegionPlot to substitute a and b for a1 and b1. I know that, for instance, Mathematica only takes less than 1 second to find the Eigenvalues of the matrix if all the parameters have a given value. But, apparently, this is not working because Mathematica doesn't apply values to a1 and b1 immediately. Unfortunately, the matrix is too complex to post it here, but if anyone has some information how to accelerate this process, that would be appreciated.

PS: Here is the matrix, depending only on the a and b values

`{{0.55 - 0.580263/(2.10526 - 1. b) - 0.025 b, 0, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), 0.55 - 0.580263/(2.10526 - 1. b) - 0.025 b, 0.1 - 0.00263158/(1.05263 - 1. b) - 0.05 b, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), (-0.102632 + (0.152632 - 0.05 b) b + a (0.00125 + (-0.00125 + 2.63678*10^-18 b) b))/(-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b), 0, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), 0, 0, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), (-0.102632 + (0.152632 - 0.05 b) b + a (0.00125 + (-0.00125 + 2.63678*10^-18 b) b))/(-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b), 0, 0}, {0, 0.025 + 0.525/(2.10526 - 1. a), ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0.025 + 0.525/(2.10526 - 1. a), ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0.05 + 0.05/(1.05263 - 1. a), ( 0.205263 - 0.0525 a - 0.05 b + 5.27356*10^-18 a b)/( 4.21053 - 2.1 a - 2. b + 2.10942*10^-16 a b), ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, ( 0.205263 - 0.0525 a - 0.05 b + 5.27356*10^-18 a b)/( 4.21053 - 2.1 a - 2. b + 2.10942*10^-16 a b), 0}, {0, 0, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), 0, 0, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), (-0.102632 + (0.152632 - 0.05 b) b + a (0.00125 + (-0.00125 + 2.63678*10^-18 b) b))/(-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b), 0, (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), 0, (-0.205263 + (0.255263 - 0.05 b) b + a (0.0525 + (-0.0525 + 5.27356*10^-18 b) b))/(-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b), (-0.340132 + (0.365132 - 0.025 b) b + a (0.013125 + (-0.013125 + 1.31839*10^-18 b) b))/(-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b), (-0.102632 + (0.152632 - 0.05 b) b + a (0.00125 + (-0.00125 + 2.63678*10^-18 b) b))/(-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b), (-0.205263 + (0.255263 - 0.05 b) b + a (0.0525 + (-0.0525 + 5.27356*10^-18 b) b))/(-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b), (-0.102632 + (0.152632 - 0.05 b) b + a (0.0025 + (-0.0025 + 5.27356*10^-18 b) b))/(-1.05263 + 0.05 a + 1. b - 1.05471*10^-16 a b)}, {0.025 + 0.525/(2.10526 - 1. b), 0, ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0.025 + 0.525/(2.10526 - 1. b), 0.05 + 0.05/(1.05263 - 1. b), ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), ( 0.102632 - 0.00125 a - 0.05 b + 2.63678*10^-18 a b)/( 2.10526 - 0.05 a - 2. b + 1.05471*10^-16 a b), 0, ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, ( 0.340132 - 0.013125 a - 0.025 b + 1.31839*10^-18 a b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), ( 0.102632 - 0.00125 a - 0.05 b + 2.63678*10^-18 a b)/( 2.10526 - 0.05 a - 2. b + 1.05471*10^-16 a b), 0, 0}, {0.475 - 0.525/(2.10526 - 1. b), 0, (0.2375 - 0.2375 b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0.475 - 0.525/(2.10526 - 1. b), 0.95 - 0.05/(1.05263 - 1. b), ( 0.2375 - 0.2375 b)/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), ( 0.475 - 0.475 b)/(2.10526 - 0.05 a - 2. b + 1.05471*10^-16 a b), 0, (0.2375 - 0.2375 b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, ( 0.2375 - 0.2375 b)/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), ( 0.475 - 0.475 b)/(2.10526 - 0.05 a - 2. b + 1.05471*10^-16 a b), 0, 0}, {0, 0, 0.2375/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, 0.2375/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, 0.2375/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, 0.2375/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, 0}, {0, 0, (0.2375 - 0.2375 b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, ( 0.2375 - 0.2375 b)/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), ( 0.475 - 0.475 b)/(2.10526 - 0.05 a - 2. b + 1.05471*10^-16 a b), 0, (0.2375 - 0.2375 b)/( 4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), 0, 0, ( 0.2375 - 0.2375 b)/(4.21053 - 1.05 a - 2. b + 1.05471*10^-16 a b), ( 0.475 - 0.475 b)/(2.10526 - 0.05 a - 2. b + 1.05471*10^-16 a b), 0, 0}, {0, 0.55 - 0.580263/(2.10526 - 1. a) - 0.025 a, ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0.55 - 0.580263/(2.10526 - 1. a) - 0.025 a, ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0.1 - 0.00263158/(1.05263 - 1. a) - 0.05 a, ( 0.106906 + (-0.156043 + 0.05 b) b + a (-0.159195 + (0.208041 - 0.05 b) b) + a^2 (0.0527916 + (-0.0525 + 5.27356*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.106906 + (-0.156043 + 0.05 b) b + a (-0.159195 + (0.208041 - 0.05 b) b) + a^2 (0.0527916 + (-0.0525 + 5.27356*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), 0}, {0, 0, (0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, 0}, {0, 0.475 - 0.525/(2.10526 - 1. a), ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0.475 - 0.525/(2.10526 - 1. a), ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0.95 - 0.05/(1.05263 - 1. a), ( 0.0100503 + a (-0.00527638 + 5.30006*10^-19 b) - 0.00477387 b)/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.0100503 + a (-0.00527638 + 5.30006*10^-19 b) - 0.00477387 b)/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), 0}, {0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.95 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.95 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), 0}, {0, 0, ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.05291 + (-0.102896 + 0.05 b) b + a (-0.0539091 + (0.103895 - 0.05 b) b) + a^2 (0.00125033 + (-0.00125 + 2.63678*10^-18 b) b))/((-1.00026 + 1. b) (-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.106906 + (-0.156043 + 0.05 b) b + a (-0.159195 + (0.208041 - 0.05 b) b) + a^2 (0.0527916 + (-0.0525 + 5.27356*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.0559654 + (-0.0792152 + 0.025 b) b + a (-0.0677186 + (0.0908954 - 0.025 b) b) + a^2 (0.0131979 + (-0.013125 + 1.31839*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.05291 + (-0.102896 + 0.05 b) b + a (-0.0539091 + (0.103895 - 0.05 b) b) + a^2 (0.00125033 + (-0.00125 + 2.63678*10^-18 b) b))/((-1.00026 + 1. b) (-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.106906 + (-0.156043 + 0.05 b) b + a (-0.159195 + (0.208041 - 0.05 b) b) + a^2 (0.0527916 + (-0.0525 + 5.27356*10^-18 b) b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.05291 + (-0.102896 + 0.05 b) b + a (-0.0551594 + (0.105145 - 0.05 b) b) + a^2 (0.00250066 + (-0.0025 + 5.27356*10^-18 b) b))/((-1.00026 + 1. b) (-1.05263 + 0.05 a + 1. b - 1.05471*10^-16 a b))}, {0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.475 (-1. + b) (-1.05263 + a (0.0526316 - 1.11022*10^-16 b) + 1. b))/((-1.00026 + 1. b) (-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.475 (-1. + b) (-1.05263 + a (0.0526316 - 1.11022*10^-16 b) + 1. b))/((-1.00026 + 1. b) (-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0}, {0, 0, ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.0100503 + a (-0.00527638 + 5.30006*10^-19 b) - 0.00477387 b)/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.00251256 + a (-0.0013191 + 1.32501*10^-19 b) - 0.00119347 b)/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.0100503 + a (-0.00527638 + 5.30006*10^-19 b) - 0.00477387 b)/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), 0}, {0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.475 (-1. + b) (-1.05263 + a (0.0526316 - 1.11022*10^-16 b) + 1. b))/((-1.00026 + 1. b) (-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), 0, ( 0.95 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.2375 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 1.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.475 (-1. + b) (-1.05263 + a (0.0526316 - 1.11022*10^-16 b) + 1. b))/((-1.00026 + 1. b) (-2.10526 + 0.05 a + 2. b - 1.05471*10^-16 a b)), ( 0.95 (-1. + b) (-2.10526 + a (1.10526 - 1.11022*10^-16 b) + 1. b))/((-1.00555 + 1. b) (-4.21053 + 2.1 a + 2. b - 2.10942*10^-16 a b)), ( 0.95 (-1. + b) (-1.05263 + a (0.0526316 - 1.11022*10^-16 b) + 1. b))/((-1.00026 + 1. b) (-1.05263 + 0.05 a + 1. b - 1.05471*10^-16 a b))}}`