Mathematica Asked on June 5, 2021
It is known that quadratic form $fleft(x_{1}, x_{2}right)=x_{1}^{2}-4 x_{1} x_{2}+4 x_{2}^{2}$ can be transformed into quadratic form $gleft(y_{1}, y_{2}right)=a y_{1}^{2}+4 x_{1} x_{2}+6 y_{2}^{2}$ by orthogonal transformation (It is known that matrix $left(begin{array}{ll}
a & 2
2 & b
end{array}right)$ and matrix $left(begin{array}{ll}
1 & -2
-2 & 4
end{array}right)$ are congruent matrices under orthogonal transformation).
Now I want to find the value of a
, b
.
Solve[Det[{{a, 2}, {2, b}}] == Det[{{1, -2}, {-2, 4}}] &&
MatrixRank[{{a, 2}, {2, b}}] == MatrixRank[{{1, -2}, {-2, 4}}], {a,
b}]
But the above code returns an empty set after being run (the answer is {a->4,b->1}
). What can I do to solve this matrix equation?
Updated content:
In addition, for the three-dimensional case, how to find the solution of the following matrix equation quickly:
Q = Array[x, {3, 3}];
A = {{1 - a, 1 + a, 0}, {1 + a, 1 - a, 0}, {0, 0, 2}} /. a -> 2;
FindInstance[
Thread[Transpose[Q] . A . Q == {{-4, 0, 0}, {0, 2, 0}, {0, 0, 2}}],
Flatten[Q], Reals]
Since Q
is required to be a real matrix, the above code has been running and cannot return results.
Other examples for testing:
A = {{a, 0, 1}, {0, a, -1}, {1, -1, a - 1}};
Transpose[Q] . A . Q == {{1, 0, 0}, {0, 1, 0}, {0, 0, 0}}
Not too hard to do, if you recall that one can use a rotation matrix to perform the required orthogonal similarity transformation:
{{{a, 2}, {2, b}}, TrigExpand[RotationMatrix[2 ArcTan[u]]]} /.
Solve[With[{rot = TrigExpand[RotationMatrix[2 ArcTan[u]]]},
Flatten[Thread /@ Thread[rot.{{a, 2}, {2, b}}.Transpose[rot] ==
{{1, -2}, {-2, 4}}]]],
{a, b, u}]
{{{{1, 2}, {2, 4}}, {{-(3/5), 4/5}, {-(4/5), -(3/5)}}},
{{{4, 2}, {2, 1}}, {{0, 1}, {-1, 0}}},
{{{1, 2}, {2, 4}}, {{3/5, -(4/5)}, {4/5, 3/5}}},
{{{4, 2}, {2, 1}}, {{0, -1}, {1, 0}}}}
Note also the use of the Weierstrass substitution to ease the algebra done by Solve[]
.
As an example verification,
{{0, -1}, {1, 0}}.{{4, 2}, {2, 1}}.Transpose[{{0, -1}, {1, 0}}] == {{1, -2}, {-2, 4}}
True
Correct answer by J. M.'s ennui on June 5, 2021
Get help from others!
Recent Answers
Recent Questions
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP