grandes-ecoles 2011 QIII.A.5

grandes-ecoles · France · centrale-maths2__mp Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$
Demonstrate that $H_n$ admits $n$ real eigenvalues (counted with their multiplicity) that are strictly positive.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by:
$$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$

Demonstrate that $H_n$ admits $n$ real eigenvalues (counted with their multiplicity) that are strictly positive.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C QI.D QII.A QII.B QII.C QII.D QII.E QII.F QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.A.5 QIV.A.6 QIV.B.1 QIV.B.2 QIV.B.3