George Herriman's Krazy Kat is lounging under an umbrella.

Linear Algebra Calculators

Jacobi Algorithm

This calculator runs the Jacobi algorithm on a symmetric matrix $A$ . This is a toy version of the algorithm and is provided solely for entertainment value.

We're looking for orthogonal $Q$ and diagonal $Lambda$ such that $AQ=Q Lambda$ . The calculator proceeds one step at a time so that the (hoped for) convergence can be watched.

The equation $AQ=Q B$ is always satisfied, and the matrix $Q$ is always orthogonal. If things go well, $B$ will converge to a diagonal matrix $Lambda.$ At this point $B$ will contain the eigenvalues of $A$ on its diagonal, while the corresponding eigenvectors of $A$ are stored in the columns of the current $Q.$

At each step we either perform a Jacobi rotation about the provided positions, or we do a sweep and perform Jacobi rotations (in sequence) for each pair of positions in the matrix. A Jacobi rotation about the positions $i$ and $j$ will set the entries $B_{ij}=B_{ji}$ to zero at the cost of possibly destroying any zeros that were already in $B.$

Either choose a size and press this button to get a randomly generated matrix, or enter your matrix in the box below. (Look at the example to see the format.)

Matrix $A$ :

`A=((0.527869806653,-0.191549824678,0.173885287532),(-0.191549824678,0.091317940076,0.461771898778),(0.173885287532,0.461771898778,0.180812253271))`

`B =((0.527869806653,-0.191549824678,0.173885287532),(-0.191549824678,0.091317940076,0.461771898778),(0.173885287532,0.461771898778,0.180812253271))`

`Q =((1,0,0),(0,1,0),(0,0,1))`

Perform a Jacobi rotation about positions and

Perform, in sequence, a rotation for each possible choice of positions.

The reset button leaves the $A$ matrix alone, but restarts the algorithm from the beginning with $B=A$ and $Q=I.$

Back to calculator page or home page.