Digits to display:	0 1 2 3 4 5 6 7 8 9
Significant digits:	1 2 3 4 5 6 7 8 machine precision
Divider after column:	{{cell.column}}
Operation type:	pivot division only (red) row subtraction only (blue, column must have a pivot selected already) forward reduction backward substitution full reduction (Gauß-Jordan method)
Show old values:	useful to extract the LU decomposition
Eigenvalue:	This value is subtracted from the diagonal.
Description:	{{ inittableau.description }}
{{ (!edit) ? "Click to edit:" : "Click to compute:" }}	{{ (!edit) ? "Edit" : "Compute" }} (Changes can be saved in the database)

	{{ (cell.column > tableau.dividercolumn) ? (cell.column - tableau.dividercolumn) : cell.column }}
{{row.row}}	{{ (showold) ? ( (cell.ispivot || cell.isrowop) ? cell.oldvalue.toFixed(digits) : cell.value.toFixed(digits) ) : ( cell.value.toFixed(digits) ) }}

	{{ (cell.column > inittableau.dividercolumn) ? (cell.column - inittableau.dividercolumn) : cell.column }}
{{row.row}}

Pivot product:

1 × {{cell.value.toFixed(digits)}}

= {{tableau.determinant().toFixed(digits)}}

{{ url + id }}

Click here to open this matrix in the Jacobi calculator.

What the calculator does

The calculator performs Gauss operations on a tableau. A Gauss operation consists of dividing a row by the pivot element and subtracting suitable multiples of the new pivot row from other rows. In the forward reduction phase of the algorithm, it's the elements below the pivot element that are reduced to zero, in the backward subsitution phase of the algorithm it's the elements above. A full reduction step does both. The pivot element is highlighted in red, the elements reduced to zero become blue.

How to use the calculator

To perform a Gauss operation, just click on the matrix element you want to use as pivot element. Backward substitution operations are only allowed if the pivot element already has value 1. No gauss operations are allowed to the right of the vertical divider line which usually stands for an equal sign.

Pivot product and determinant

When doing full reductions, the pivot product is equal to the determinant up to a sign. Forward reduction is sufficient if the standard sequence of pivot elements along the main diagonal is followed. The sign is the sign of the permutation defined by the choice of pivot elements.

Eigenvectors

The eigenvector algorithm requires that for each eigenvalue λ, the linear system of equations with matrix A-λI must be solved to find eigenvectors. Enter an eigenvalue in the Eigenvalue and click reset to reload the tableau with the matrix A-λI.