# The Unapologetic Mathematician

## Unitary and Orthogonal Matrices and Orthonormal Bases

I almost forgot to throw in this little observation about unitary and orthogonal matrices that will come in handy.

Let’s say we’ve got a unitary transformation $U$ and an orthonormal basis $\left\{e_i\right\}_{i=1}^n$. We can write down the matrix as before

$\displaystyle\begin{pmatrix}u_{1,1}&\cdots&u_{1,n}\\\vdots&\ddots&\vdots\\u_{n,1}&\cdots&u_{n,n}\end{pmatrix}$

Now, each column is a vector. In particular, it’s the result of transforming a basis vector $e_i$ by $U$.

$\displaystyle U(e_i)=u_{1,i}e_1+\dots+u_{n,i}e_n$

What do these vectors have to do with each other? Well, let’s take their inner products and find out.

$\displaystyle\langle U(e_i),U(e_j)\rangle=\langle e_i,e_j\rangle=\delta_{i,j}$

since $U$ preserves the inner product. That is the collection of columns of the matrix of $U$ form another orthonormal basis.

On the other hand, what if we have in mind some other orthonormal basis $\left\{f_j\right\}_{j=1}^n$. We can write each of these vectors out in terms of the original basis

$\displaystyle f_j=a_{1,j}e_1+\dots+a_{n,j}e_n$

and even get a change-of-basis transformation (like we did for general linear transformations) $A$ defined by

$\displaystyle A(e_j)=f_j=a_{1,j}e_1+\dots+a_{n,j}e_n$

so the $a_{i,j}$ are the matrix entries for $A$ with respect to the basis $\left\{e_i\right\}$. This transformation $A$ will then be unitary.

Indeed, take arbitrary vectors $v=v^ie_i$ and $w=w^je_j$. Their inner product is

$\displaystyle\langle v,w\rangle=\langle v^ie_i,w^je_j\rangle=\overline{v^i}w^j\langle e_i,e_j\rangle=\overline{v^i}w^j\delta_{i,j}$

On the other hand, after acting by $A$ we find

$\displaystyle\langle A(v),A(w)\rangle=\langle v^iA(e_i),w^jA(e_j)\rangle=\overline{v^i}w^j\langle f_i,f_j\rangle=\overline{v^i}w^j\delta_{i,j}$

since the basis $\left\{f_j\right\}$ is orthonormal as well.

To sum up: with respect to an orthonormal basis, the columns of a unitary matrix form another orthonormal basis. Conversely, writing any other orthonormal basis in terms of the original basis and using these coefficients as the columns of a matrix gives a unitary matrix. The same holds true for orthogonal matrices, with similar reasoning all the way through. And both of these are parallel to the situation for general linear transformations: the columns of an invertible matrix with respect to any basis form another basis, and conversely.

August 7, 2009 - Posted by | Algebra, Linear Algebra