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Matrix with given Eigenvalues
ew-01-01
multiple
9792
randFromArray([-19,-17,-13,-11,-7,-5,-3,-2,-1,19,17,13,11,7,5,3,2,1]) randFromArray([-19,-17,-13,-11,-7,-5,-3,-2,-1,19,17,13,11,7,5,3,2,1]) randRange(-8,8) randRangeNonZero(-8,8) L1+L2-A (A*D - L1 * L2)/C

Determine the entry {\color{red}b} in \begin{pmatrix} A & {\color{red}b}\\ C & D \end{pmatrix}, such that the matrix has eigenvalues \color{orange} \lambda_1 = L1 and \color{blue} \lambda_2 = L2.

b \color{red} b = LSG

We can use that the product of the eigenvalues is the determinant of the matrix.

Using this observation, we need to find {\color{red}b} such that negParens(A) \cdot negParens(D) - {\color{red}b} \cdot negParens(C) = {\color{orange}negParens(L1)} \cdot {\color{blue}negParens(L2)} = {\color{teal}L1 * L2}.

Solving for {\color{red}b} yields {\color{red}b} = \dfrac{L1 * L2 - A*D}{-C} = fractionReduce(A*D - L1 * L2,C).

Note: For the argument we need the eigenvalues to be prime numbers (or negative primes).

Otherwise the determinant coud be another produkt with \color{orange} \lambda_1 = L1 and \color{blue} \lambda_2 = L2.