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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.

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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.

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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.