5x5 Hilbert matrix - Wolfram|Alpha

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Input

HilbertMatrix[5]

Result

(1 | 1/2 | 1/3 | 1/4 | 1/5
1/2 | 1/3 | 1/4 | 1/5 | 1/6
1/3 | 1/4 | 1/5 | 1/6 | 1/7
1/4 | 1/5 | 1/6 | 1/7 | 1/8
1/5 | 1/6 | 1/7 | 1/8 | 1/9)

Dimensions

5 (rows) × 5 (columns)

Matrix plot

Matrix plot

Transpose

(1 | 1/2 | 1/3 | 1/4 | 1/5
1/2 | 1/3 | 1/4 | 1/5 | 1/6
1/3 | 1/4 | 1/5 | 1/6 | 1/7
1/4 | 1/5 | 1/6 | 1/7 | 1/8
1/5 | 1/6 | 1/7 | 1/8 | 1/9)

Property

symmetric

hankel

Trace

563/315

Determinant

1/266716800000

Inverse

5(5 | -60 | 210 | -280 | 126
-60 | 960 | -3780 | 5376 | -2520
210 | -3780 | 15876 | -23520 | 11340
-280 | 5376 | -23520 | 35840 | -17640
126 | -2520 | 11340 | -17640 | 8820)

Characteristic polynomial

-λ^5 + (563 λ^4)/315 - (735781 λ^3)/2116800 + (852401 λ^2)/222264000 - (61501 λ)/53343360000 + 1/266716800000

Characteristic polynomial »

Eigenvalues

λ_1≈1.56705

λ_2≈0.208534

λ_3≈0.0114075

λ_4≈0.000305898

λ_5≈3.28793×10^-6

Eigenvectors

v_1≈(3.65954, 2.12461, 1.53262, 1.20787, 1)

v_2≈(-1.40292, 0.643135, 0.990358, 1.03471, 1)

v_3≈(0.379009, -1.28116, -0.213119, 0.547731, 1)

v_4≈(-0.08458, 0.775946, -1.19683, -0.417906, 1)

v_5≈(0.0164091, -0.310151, 1.3453, -2.03907, 1)

Condition number

943656

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