Enter the six independent entries of a real symmetric 3×3 matrix and the three components of a vector. All inputs are dimensionless numeric scalars; negative values and decimals are allowed. The off-diagonal entries are mirrored automatically: a₂₁ = a₁₂, a₃₁ = a₁₃ and a₃₂ = a₂₃. The calculator evaluates the quadratic form and, for a nonzero vector, the Rayleigh quotient.

\[A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{12}&a_{22}&a_{23}\\a_{13}&a_{23}&a_{33}\end{pmatrix},\qquad x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\]
\[\begin{aligned}q(x)&=x^TAx=a_{11}x_1^2+a_{22}x_2^2+a_{33}x_3^2+2a_{12}x_1x_2+2a_{13}x_1x_3+2a_{23}x_2x_3,\\s(x)&=x^Tx=x_1^2+x_2^2+x_3^2,\\R_A(x)&=\frac{q(x)}{s(x)},\qquad x\ne0.\end{aligned}\]

For a real symmetric matrix, the Rayleigh quotient lies between its smallest and largest eigenvalues. If the supplied nonzero vector is an eigenvector, the quotient equals its eigenvalue. For an arbitrary direction, the result need not be an eigenvalue. Multiplying the vector by a nonzero scalar leaves the quotient unchanged, while the quadratic form is multiplied by the square of that scalar. These properties follow from the symmetric eigenvalue formulation in Cornell’s numerical analysis notes.

This calculator evaluates the supplied direction only. It does not compute eigenvectors, diagonalize the matrix, optimize the quotient or classify definiteness.

The defaults give A = [[2, 1, 0], [1, 3, 0], [0, 0, 4]] and x = (1, 2, 1). The diagonal terms sum to 2 + 12 + 4 = 18, and the off-diagonal terms contribute 4. Therefore xᵀAx = 22, xᵀx = 6 and the Rayleigh quotient is 22/6 = 11/3, approximately 3.66666667.

For x = (0, 0, 0), the quadratic form and squared vector length are both zero. The Rayleigh quotient is undefined, so division by zero produces an unavailable result rather than zero or infinity.

Calculations use finite-precision arithmetic. Very large or very small inputs can exceed the supported numeric range or lose precision; cancellation can also reduce accuracy. A numeric-range failure makes affected results unavailable, and dependent results remain unavailable. An unavailable result must not be interpreted as zero. Displayed decimals are rounded.

Reference

  1. David Bindel, CS 4220: Numerical Analysis — Applications of eigenvalue problems, “The symmetric case,” Cornell University, March 2, 2026.