hi, guys, how to understand the math used in CDXUTDirectionWidget ::UpdateLightDir 

the  following code snippet is taken from MS DXTU source code

  D3DXMATRIX mInvView;
    D3DXMatrixInverse( &mInvView, NULL, &m_mView );
    mInvView._41 = mInvView._42 = mInvView._43 = 0;

    D3DXMATRIX mLastRotInv;
    D3DXMatrixInverse( &mLastRotInv, NULL, &m_mRotSnapshot );

    D3DXMATRIX mRot = *m_ArcBall.GetRotationMatrix();
    m_mRotSnapshot = mRot;

    // Accumulate the delta of the arcball's rotation in view space.
    // Note that per-frame delta rotations could be problematic over long periods of time.
    m_mRot *= m_mView * mLastRotInv * mRot * mInvView;

    // Since we're accumulating delta rotations, we need to orthonormalize 
    // the matrix to prevent eventual matrix skew
    D3DXVECTOR3* pXBasis = ( D3DXVECTOR3* )&m_mRot._11;
    D3DXVECTOR3* pYBasis = ( D3DXVECTOR3* )&m_mRot._21;
    D3DXVECTOR3* pZBasis = ( D3DXVECTOR3* )&m_mRot._31;
    D3DXVec3Normalize( pXBasis, pXBasis );
    D3DXVec3Cross( pYBasis, pZBasis, pXBasis );
    D3DXVec3Normalize( pYBasis, pYBasis );
    D3DXVec3Cross( pZBasis, pXBasis, pYBasis );
 

https://github.com/Microsoft/DXUT/blob/master/Optional/DXUTcamera.cpp

Probably easier to ask what you don't understand about it.

Do you understand what the basis vectors are?  Do you understand what the cross product operations do? What about the function is confusing to you?