Largest root of polynomial / formula for polynomial

Question

Consider the polynomial $$ x^3-x^2-1. $$ Its largest root is $$ x_0\approx 1.466. $$ Moreover, $x^3-x^2-1$ is the characteristic polynomial of the matrix $$ A=\begin{pmatrix}1 & 1 & 0\\0 & 0 & 1\\1 & 0 & 0\end{pmatrix}. $$

Next, consider $x_0^2\approx 1.466^2$. This is the largest root of $$ x^3-x^2-2x-1 $$ which is the characteristic polynomial of the matrix $$ \begin{pmatrix}1 & 1 & 1\\1 & 0 & 0\\1 & 1 & 0\end{pmatrix}=A^2. $$

Next $x_0^3$ which is the largest root of $$ x^3-4x^2+3x-1 $$ which is the characteristic polynomial of $$ \begin{pmatrix}2 & 1 & 1\\1 & 1 & 0\\1 & 1 & 1\end{pmatrix}=A^3 $$

I think that, in general, for $a\geq 1$, we have that $x_0^a$ is the largest root of the characteristic polynomial of the matrix $A^a$.

Correct?

Answer 1

I don't know if that helps but here are the $32$ possible polynomials for $3\times 3$ matrix with $\{0,1\}$ entries.

The numbers on the right are binaries entries in flat mode. For instance $140=010001100$ will stand for matrix $((0,1,0),(0,0,1),(1,0,0))$.

x^3             : 0,2,4,6,8,12,32,34,36,38,40,44,64,66,72,96,104,128,130,132,134,136,192,194,200
x^3-1           : 98,140
x^3-2*x         : 78,170,228
x^3-2*x^2       : 27,31,59,63,91,119,155,189,219,221,243,325,327,333,357,363,365,374,378,399,414,429,432,434,436,438,440,453,455,476,483,496,504
x^3-2*x^2+1     : 85,87,93,117,125,177,179,181,183,185,213,215,241,249,267,271,282,286,299,303,314,318,331,340,342,346,348,372,380,395,410,417,419,421,423,425,459,468,470,474,481,489
x^3-2*x^2+x     : 17,19,21,23,25,29,49,51,53,55,57,61,81,83,89,113,121,145,147,149,151,153,209,211,217,257,259,261,263,265,269,272,274,276,278,280,284,289,291,293,295,297,301,304,306,308,310,312,316,321,323,329,336,338,344,353,361,368,376,385,387,389,391,393,400,402,404,406,408,449,451,457,464,466,472
x^3-2*x^2+x-1   : 115,157,355,370,397,412
x^3-2*x^2-1     : 123,159,359,444,461,498
x^3-2*x^2-2*x   : 255,495,510
x^3-2*x^2-x     : 127,191,223,251,367,446,463,487,493,502,506,508
x^3-2*x^2-x+1   : 95,187,247,253,335,382,431,442,478,485,491,500
x^3-2*x^2-x+2   : 245,350,427
x^3-2*x-1       : 110,174,206,230,234,236
x^3-3*x^2       : 511
x^3-3*x^2+2*x   : 283,287,315,319,341,343,347,349,373,381,411,433,435,437,439,441,469,471,475,497,505
x^3-3*x^2+2*x-1 : 375,379,415,445,477,499
x^3-3*x^2+3*x-1 : 273,275,277,279,281,285,305,307,309,311,313,317,337,339,345,369,377,401,403,405,407,409,465,467,473
x^3-3*x^2+3*x-2 : 371,413
x^3-3*x^2+x     : 383,447,479,503,507,509
x^3-3*x^2+x+1   : 351,443,501
x^3-3*x-2       : 238
x^3-x           : 10,14,42,46,68,70,74,76,100,108,138,160,162,164,166,168,196,198,202,224,232
x^3-x^2         : 1,3,5,7,9,13,16,18,20,22,24,28,33,35,37,39,41,45,48,50,52,54,56,60,65,67,73,80,82,88,97,105,112,120,129,131,133,135,137,144,146,148,150,152,193,195,201,208,210,216,256,258,260,262,264,268,288,290,292,294,296,300,320,322,328,352,360,384,386,388,390,392,448,450,456
x^3-x^2-1       : 99,114,141,156,354,396
x^3-x^2-2*x     : 79,126,175,186,222,231,235,237,246,252,366,430,462,484,490
x^3-x^2-2*x+1   : 94,171,229,244,334,426
x^3-x^2-2*x-1   : 111,190,207,250,486,492
x^3-x^2-3*x-1   : 239,254,494
x^3-x^2-x       : 11,15,26,30,43,47,58,62,69,71,75,77,90,101,109,118,139,154,173,176,178,180,182,184,197,199,203,218,220,227,240,248,324,326,332,356,362,364,398,416,418,420,422,424,452,454,480,488
x^3-x^2-x+1     : 84,86,92,116,124,161,163,165,167,169,212,214,225,233,266,270,298,302,330,394,458
x^3-x^2-x-1     : 103,107,122,143,158,188,205,242,358,428,460,482
x^3-x-1         : 102,106,142,172,204,226