Checking the mechanical stability of input elastic constant
if we write single-crystalline elastic constant matrix in general as follows

C11 C12 C13 C14 C15 C16 C22 C23 C24 C25 C26 C33 C34 C35 C36 C44 C45 C46 C55 C56 C66

where the lower part is empty as the matrix is symmetric C_αβ = C_βα, conditions of mechanical stability require that this matrix is positive definite. This condition is fulfilled in case of real symmetric matrices if and only is the so-called leading principal minors of the matrix are positive. These are defined by the following relations (using determinants |A| of a matrix A)

Det

C11

=

C11

>

0,

Det

C11 C12 C21 C22

>

0,

Det

C11 C12 C13 C21 C22 C23 C31 C32 C33

>

0,

Det

C11 C12 C13 C14 C21 C22 C23 C24 C31 C32 C33 C34 C41 C42 C43 C44

>

0,

Det

C11 C12 C13 C14 C15 C21 C22 C23 C24 C25 C31 C32 C33 C34 C35 C41 C42 C43 C44 C45 C51 C52 C53 C54 C55

>

0,

Det

C11 C12 C13 C14 C15 C16 C21 C22 C23 C24 C25 C26 C31 C32 C33 C34 C35 C36 C41 C42 C43 C44 C45 C46 C51 C52 C53 C54 C55 C56 C16 C26 C36 C46 C56 C66

>

0,

Specifically in case of crystals with a cubic symmetry, these conditions can be equivalently written as

C₁₁ + 2C₁₂  > 0,

C₄₄  > 0,

C₁₁ - C₁₂  > 0.