# Thread: Orthonormal change of basis matrix

1. ## Orthonormal change of basis matrix

Suppose I have a matrix $A$ w.r.t some basis $v_1,....,v_n$.

Further suppose that this matrix undergoes a basis change to $\underline{v}_1,...., \underline{v}_n$ where each $\underline{v}_i$ is orthonormal to another (ie. $\underline{v}_1,...., \underline{v}_n$ is an orthonormal basis). Call this matrix $B$ (which has columns $\underline{v}_1,...., \underline{v}_n$).

Let the change of basis matrix be $Q$.

Is it true that $A=Q^T B Q$?

Is it also true that $det(Q)=1$?

(P.S I've done a question and this is one of the facts I used. I'm wondering if it really is a fact!).

2. The change of basis matrix is orthogonal, which implies that its determinant is $\pm1$. It will be +1 if the basis change is orientation-preserving, and –1 if it is orientation-reversing. The formula $A = Q^{\textsc t}BQ$ is correct.