# How do I solve this matrix equation with positive definite matrix constraints? [closed]

I need to determine the range of a that makes the iteration formula of matrix $$\left(\begin{array}{lll} 1 & a & a \\ a & 1 & a \\ a & a & 1 \end{array}\right)$$ converge when using Jacobian iteration method $$\color{Gray} {\text{(2001 武汉 岩石 数值分析 7)}}$$.

Reduce[And @@ Thread[Abs[Eigenvalues[-{{0, a, a}, {a, 0, a}, {a, a, 0}}]] < {1, 1, 1}], Reals]


In fact, this problem can also be solved by the following method:

Reduce[PositiveDefiniteMatrixQ[{{1, a, a}, {a, 1, a}, {a, a, 1}}] ==
True &&
PositiveDefiniteMatrixQ[
2*IdentityMatrix[3] - {{1, a, a}, {a, 1, a}, {a, a, 1}}] ==
True, a]


I want to know how to solve the above matrix equation?

The results of the following methods are correct, but the process is not exquisite and does not meet the requirements:

Reduce[Det[{{1, a}, {a, 1}}] > 0 &&
Det[{{1, a, a}, {a, 1, a}, {a, a, 1}}] > 0 &&
Det[2*IdentityMatrix[3] - {{1, a, a}, {a, 1, a}, {a, a, 1}}] >
0 &&
Det[{{1, -a}, {-a, 1}}] > 0, a]

• Is your question about how to change Abs[x] < 1/2 into -1/2 < x < 1/2? If so you might try with pattern matching: Abs[x] < 1/2 /. Abs[x_] < y_ :> -y < x < y. – anderstood Sep 15 at 7:38
• Once again a question about the underlying math rather than the software Mathematica. Such questions should be directed to an appropriate forum, which this is not. – Daniel Lichtblau Sep 15 at 13:32

m = {{1, a, a}, {a, 1, a}, {a, a, 1}}