Numerical Instability
Linear algebra properties describe systems of linear equations where small perturbations in input data cause large variations in computed solutions. Severe matrix ill conditioning occurs in multi-axis sensor calibration matrices when sensing axes lack orthogonality or share near-collinear responses. Inverting an ill-conditioned coefficient matrix amplifies measurement noise into severe calibration errors.
Sensor array designs require distinct orthogonal spatial responses to prevent ill-conditioned mathematical transformations.
Condition Number
Ratio of largest to smallest singular values defines the condition number of a calibration matrix. Large condition numbers indicate near-singular matrices that lose numerical precision during inversion operations. Floating-point roundoff errors during matrix inversion corrupt calculated cross-axis sensitivity coefficients.
Error Amplification
Noise on raw sensor channels translates directly into erratic physical vector outputs. In multi-axis accelerometer matrices, minor voltage noise on one channel causes large false motion vectors on orthogonal axes. Mechanical axis alignment minimizes cross-axis sensitivity prior to software matrix inversion.
Regularization Boundary
Tikhonov regularization and singular value truncation mitigate numerical instability by constraining inversion gains. Introducing small penalty terms stabilizes matrix ill conditioning at the cost of slight systematic bias in calculated outputs. System calibration routines flag matrix condition numbers exceeding one hundred as invalid configuration setups.
