grandes-ecoles 2017 QV.B.4

grandes-ecoles · France · centrale-maths2__psi Matrices Matrix Decomposition and Factorization

We fix a natural number $p$ greater than or equal to 2. Prove that every invertible matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ admits at least one $p$-th root.
One may use the fact that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root (as established in V.B.3).

We fix a natural number $p$ greater than or equal to 2. Prove that every invertible matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ admits at least one $p$-th root.

One may use the fact that every upper triangular and invertible matrix admits at least one upper triangular $p$-th root (as established in V.B.3).

Paper Questions

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