grandes-ecoles 2013 QII.A.1

grandes-ecoles · France · centrale-maths2__mp Matrices Matrix Algebra and Product Properties

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
Justify that ${}^t A = U({}^t B)U^{-1}$.

In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.

Justify that ${}^t A = U({}^t B)U^{-1}$.

Paper Questions

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