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.

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

3 Comments »

[...] Now if we have an arbitrary orthonormal basis — say is a transformation on with the standard basis already floating around — we may want to work with the matrix of with respect to this basis. If this were our basis of eigenvectors, would have the diagonal matrix . But we may not be so lucky. Still, we can perform a change of basis using the basis of eigenvectors to fill in the columns of the change-of-basis matrix. And since we’re going from one orthonormal basis to another, this will be unitary! [...]

Pingback by The Complex Spectral Theorem « The Unapologetic Mathematician | August 10, 2009 | Reply
[...] already floating around for , we can use this new basis to perform a change of basis, which will be orthogonal (not unitary in this case). That is, we can write the matrix of any self-adjoint transformation as [...]

Pingback by The Real Spectral Theorem « The Unapologetic Mathematician | August 14, 2009 | Reply
[...] numbers) we find that they’re orthonormal. But this means that the modified table is a unitary matrix, and thus its columns are orthonormal as well. We conclude [...]

Pingback by The Character Table as Change of Basis « The Unapologetic Mathematician | November 22, 2010 | Reply

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