3 added 99 characters in body edited Sep 24 '14 at 1:23 DumpsterDoofus 9,7812020 silver badges4545 bronze badges The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}} - 2 IdentityMatrix[2]] giving {1/2 (-42 + t11^2 + t12^2 + t22^2 - Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4]), 1/2 (-42 + t11^2 + t12^2 + t22^2 + Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to (-2 + t11^2 + t12^2 + t22^2)^2 > t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4 $$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2-4> \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$ which which simplifies to $$t_{11}^2 t_{22}^2>1$$$$1 + t11^2 (-1 + t22^2) > t12^2 + t22^2$$ which is automatically satisfied by your entry conditionscan be false when t12 becomes arbitrarily large. Thus the matrix is not always positive-definite definite; Daniel Lichtblau's answer shows a way to construct explicit counterexamples. The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}} - 2 IdentityMatrix[2]] giving {1/2 (-4 + t11^2 + t12^2 + t22^2 - Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4]), 1/2 (-4 + t11^2 + t12^2 + t22^2 + Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to $$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2-4> \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$ which simplifies to $$t_{11}^2 t_{22}^2>1$$ which is automatically satisfied by your entry conditions. Thus the matrix is positive-definite. The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}} - IdentityMatrix[2]] giving {1/2 (-2 + t11^2 + t12^2 + t22^2 - Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4]), 1/2 (-2 + t11^2 + t12^2 + t22^2 + Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to (-2 + t11^2 + t12^2 + t22^2)^2 > t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4 which simplifies to $$1 + t11^2 (-1 + t22^2) > t12^2 + t22^2$$ which can be false when t12 becomes arbitrarily large. Thus the matrix is not always positive definite; Daniel Lichtblau's answer shows a way to construct explicit counterexamples. 2 added 97 characters in body edited Sep 23 '14 at 1:28 DumpsterDoofus 9,7812020 silver badges4545 bronze badges The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}}] - 2 IdentityMatrix[2]] giving {1/2 (t11^2-4 + t11^2 + t12^2 + t22^2 - Sqrt[-4 Sqrt[ t11^4 + 2 t11^2 t22^2t12^2 + (-t11^2t12^4 - 2 t11^2 t22^2 + 2 t12^2 - t22^2)^2] + t22^4]), 1/2 (t11^2-4 + t11^2 + t12^2 + t22^2 + Sqrt[-4Sqrt[ t11^4 + 2 t11^2 t22^2t12^2 + (-t11^2t12^4 - 2 t11^2 t22^2 + 2 t12^2 - t22^2)^2] + t22^4])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to $$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2\geq \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$$$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2-4> \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$ which simplifies to $$t_{11}^2 t_{22}^2\geq 0$$$$t_{11}^2 t_{22}^2>1$$ which is automatically satisfied by your entry conditions. Thus the matrix is positive-definite. The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}}] giving {1/2 (t11^2 + t12^2 + t22^2 - Sqrt[-4 t11^2 t22^2 + (-t11^2 - t12^2 - t22^2)^2]), 1/2 (t11^2 + t12^2 + t22^2 + Sqrt[-4 t11^2 t22^2 + (-t11^2 - t12^2 - t22^2)^2])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to $$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2\geq \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$ which simplifies to $$t_{11}^2 t_{22}^2\geq 0$$ which is automatically satisfied by your entry conditions. Thus the matrix is positive-definite. The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}} - 2 IdentityMatrix[2]] giving {1/2 (-4 + t11^2 + t12^2 + t22^2 - Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4]), 1/2 (-4 + t11^2 + t12^2 + t22^2 + Sqrt[ t11^4 + 2 t11^2 t12^2 + t12^4 - 2 t11^2 t22^2 + 2 t12^2 t22^2 + t22^4])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to $$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2-4> \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$ which simplifies to $$t_{11}^2 t_{22}^2>1$$ which is automatically satisfied by your entry conditions. Thus the matrix is positive-definite. 1 answered Sep 23 '14 at 1:18 DumpsterDoofus 9,7812020 silver badges4545 bronze badges The eigenvalues are Eigenvalues[{{t11^2, t11*t12}, {t11*t12, t12^2 + t22^2}}] giving {1/2 (t11^2 + t12^2 + t22^2 - Sqrt[-4 t11^2 t22^2 + (-t11^2 - t12^2 - t22^2)^2]), 1/2 (t11^2 + t12^2 + t22^2 + Sqrt[-4 t11^2 t22^2 + (-t11^2 - t12^2 - t22^2)^2])} Assuming that the radicand is real-valued, one then notes that positive-definiteness is equivalent to $$\left(t_{11}^2+t_{12}^2+t_{22}^2\right){}^2\geq \left(-t_{11}^2-t_{12}^2-t_{22}^2\right){}^2-4 t_{11}^2 t_{22}^2$$ which simplifies to $$t_{11}^2 t_{22}^2\geq 0$$ which is automatically satisfied by your entry conditions. Thus the matrix is positive-definite.