Stamp of Hilbert

In Linear algebra, the matrix of Hilbert is a square Matrice of general term B ij = 1/( I + J − 1). It is named thus in homage to the mathematician David Hilbert. The matrices of Hilbert are used as traditional examples of badly conditioned matrices , which makes from there the use very delicate in numerical Analyze. For example, the coefficient of conditioning (for standard 2) of the matrix which follows is about 4.8 · 105.

Thus the matrix of Hilbert of size 5 is worth

B = \ begin {bmatrix}
1 & \ frac {1} {2} & \ frac {1} {3} & \ frac {1} {4} & \ frac {1} {5} \ \ \ frac {1} {2} & \ frac {1} {3} & \ frac {1} {4} & \ frac {1} {5} & \ frac {1} {6} \ \ \ frac {1} {3} & \ frac {1} {4} & \ frac {1} {5} & \ frac {1} {6} & \ frac {1} {7} \ \ \ frac {1} {4} & \ frac {1} {5} & \ frac {1} {6} & \ frac {1} {7} & \ frac {1} {8} \ \ \ frac {1} {5} & \ frac {1} {6} & \ frac {1} {7} & \ frac {1} {8} & \ frac {1} {9} \ end {bmatrix}.

The determinant of this matrix can be calculated explicitly, like particular case of a Déterminant of Cauchy.

If one interprets the general term of the matrix of Hilbert like

B_ {ij} = \ int_ {0} ^ {1} x^ {i+j-2} \, dx,
one can recognize there a matrix of Gram for the functions powers and the adapted scalar product.

The matrices of Hilbert are definite positive.

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