Question

(a) If Cis orthogonal and Mis symmetric, show that ${\mathit{C}}^{\mathbf{-}\mathbf{1}}\mathit{M}\mathit{C}$is symmetric.

(b) IfC is orthogonal and Mantisymmetric, show that${\mathit{C}}^{\mathbf{-}\mathbf{1}}\mathit{M}\mathit{C}$is antisymmetric.

Short answer

a)$C^{-1}MC$is symmetric.

b) $C^{-1}MC$is antisymmetric.

Step-by-step solution

Given information from question

a) It is given that, C is orthogonal, and M is symmetric.

b) It is given that, C is orthogonal, and M is antisymmetric.

Orthogonal matrix

An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix with orthonormal vectors in the columns and rows.

Calculate C-1MC is symmetric

a)

We have givenis orthogonal andis symmetric.

So,

$$\begin{gathered} {\left(C^{-1}MC\right)}^T={\left(C^T\right)}^{-1}{M}^T{C}^T \\ ={\left(C^{-1}\right)}^{-1}{M}^T{C}^T \\ =CM{C}^{-1} \end{gathered}$$

Where $C^T=C^{-1}$since C is orthogonal. And $M^T=M$where M is symmetric

Calculate C-1MC is antisymmetric

$M^T=M$(b)

Since we have givenis orthogonal andis antisymmetric.

So,

$$\begin{gathered} {\left(C^{-1}MC\right)}^T={\left(C^T\right)}^{-1}{M}^T{C}^T \\ ={\left(C^{-1}\right)}^{-1}{M}^T{C}^T \\ =C\left(-M\right)C^{-1} \\ =-CM{C}^{-1} \end{gathered}$$

Where $C^T=C^{-1}$since C is orthogonal and $M^T=M$where M is antisymmetric.