Prove that every unitary matrix is diagonalisable by a unitary matrix

In summary: All you had to do was show that every element in ##C C^{*}## is orthogonal to every other element and that the norm of each element is 1.
  • #1
Hall
351
88
Homework Statement
The same as the title.
Relevant Equations
A matrix is said to be unitary if A*A = AA* = I. A

A matrix ##A## is diagonalisable if there is a matrix C such that
##\Lambda = C^{-1} A C ##.
Let's assume that ##A## is unitary and diagonalisable, so, we have
## \Lambda = C^{-1} A C ##

Since, ##\Lambda## is made up of eigenvalues of ##A##, which is unitary, we have ## \Lambda \Lambda^* = \Lambda \bar{\Lambda} = I##.

I tried using some, petty, algebra to prove that ##C C* = I## but was unsuccessful. Here is my attempt:
## \Lambda = C^{-1} A C ##
## \Lambda^* = C^* A^* (C^*)^{-1} ##
## \Lambda \Lambda^* = C ^{-1} A C ~ C^* A^* (C^{-1})^{*} ##
## I = C ^{-1} A C ~ C^* A^* (C^{-1})^{*} ##
## I = C^{-1} AC ~ [ C^{-1} A C ]^{*} ##
That implies,
## [ C^{-1} A C ]^{*} = [C^{-1} A C]^{-1} ##
## C^* A^* (C^{-1})^{*} = C^{-1} A^* C##, since ##A^* = A^{-1}##.
But I cannot conclude ##C^* =C^{-1}##.
 
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  • #2
The question doesn't imply that a unitary matrix is only diagonalisable by a unitary transformation matrix. In fact, you should be able to see why that cannot be the case.

You cannot, therefore, prove that ##C## is unitary. Instead, you must show that you can find a unitary matrix ##C## that does the job.
 
  • #3
If ##C = [u_1 ~ \cdots u_n]##, where ##u_i## is normalised eigenvector of A, then we can show that ##C^{-1} A C## will be a diagonal matrix, made up of eigenvalues of A.
But how do we show that CC* = I. We have only two facts with us, that every ##u_i## is orthogonal to every other ##u_j## and norm of each ##u_i## is 1.

So, we have
$$
C^{*}=
\begin{bmatrix}
\overline{u_1} \\
\vdots\\
\overline{u_n}\\
\end{bmatrix}
$$

$$
CC^{*}= u_1 \overline{u_1} + \cdots u_n \overline{u_n} $$

I really have no idea of what to do next.
 
  • #4
Hall said:
If ##C = [u_1 ~ \cdots u_n]##, where ##u_i## is normalised eigenvector of A, then we can show that ##C^{-1} A C## will be a diagonal matrix, made up of eigenvalues of A.
But how do we show that CC* = I. We have only two facts with us, that every ##u_i## is orthogonal to every other ##u_j## and norm of each ##u_i## is 1.

So, we have
$$
C^{*}=
\begin{bmatrix}
\overline{u_1} \\
\vdots\\
\overline{u_n}\\
\end{bmatrix}
$$

$$
CC^{*}= u_1 \overline{u_1} + \cdots u_n \overline{u_n} $$

I really have no idea of what to do next.
You seem to be confusing matrix multiplication with matrix element multiplication.

What can you say about the eigenvalues of a unitary matrix?

If you can prove that a unitary matrix has a set of mutually orthonormal eigenvectors (or assume this), then you should be able to complete the proof using what you have.
 
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  • #5
Well, I can write C and C* in expanded form to avoid confusion:
$$
C =
\begin{bmatrix}
u_{11} & u_{12} & \cdots & u_{1n} \\
u_{21} & u_{22} & \cdots & u_{2n} \\
\vdots & \vdots & \cdots & \vdots \\
u_{n1} & u_{n2} & \cdots & \ u_{nn} \\
\end{bmatrix}
$$
Where the first column is the first normalised eigenvector, and so on.

$$
C^{*} =
\begin{bmatrix}
\overline{u_{11} } & \overline{ u_{21} } & \cdots & \overline{ u_{n1} } \\
\overline{ u_{12} } & \overline{ u_{22} } & \cdots & \overline{ u_{n2} } \\
\vdots \\
\overline{ u_{1n} } & \overline{ u_ {2n} } & \cdots & \overline{ u_{nn} } \\
\end{bmatrix}
$$

The ij th entry of CC* is
$$
CC^{*}_{ij} = [ u_{i1} ~u_{i2} \cdots u_{in} ] \times \begin{bmatrix} \overline{ u_{j1} } \\ \overline{ u_{j2} } \\ \vdots \\ \overline { u_{jn} } \\ \end{bmatrix}
$$

$$CC^{*}_{ij} = \sum u_{ik}~ \overline{ u_{jk} } $$

Then?
 
  • #6
It might be clearer if you looked at ##C^*C##. Also, you should have complex conjugates of the elements in ##C^*##.
 
  • #7
Not getting your hint. I have shown my attempts.
 
  • #8
Hall said:
Not getting your hint. I have shown my attempts.
You must learn to recognise an inner product when you see one!
 
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  • #9
Inner product: ##\langle u_i, u_j \rangle = 0##.
 
  • #11
It is easy to show ##C^{*} C = I##, that is:
$$
C_{\iota}^{*}=
\begin{bmatrix}
\overline{ u_{1\iota} } & \overline{ u_{2\iota} }& \cdots \overline{n \iota}
\end{bmatrix}

\times\begin{bmatrix}
u_{1j} \\
u_{2j}\\
\vdots \\
u_{nj} \\
\end{bmatrix}= C^j
$$

$$
(C^{*} C)_{\iota ~j} = u_{1j} \overline{ u_{1\iota}} + u_{2j} \overline{ u_{2\iota}} + \cdots + u_{nj} \overline{ u_ { n \iota} } = u_i \cdot \overline{u_j} = \langle u_i, u_j \rangle = 0 $$

Thus, all ij entry of ##C^{*}C## is zero, and by taking ##\iota= j## in the above formula I will get ##\iota- \iota## entry as 1.

So, ## C^{*} C = I##

I don't understand why ##C C^{*} = I## was so hard to prove.
 

FAQ: Prove that every unitary matrix is diagonalisable by a unitary matrix

1. What is a unitary matrix?

A unitary matrix is a complex square matrix whose conjugate transpose is equal to its inverse. In other words, it satisfies the equation U*U-1 = U-1*U = I, where U* is the conjugate transpose of U and I is the identity matrix.

2. What does it mean for a matrix to be diagonalisable by a unitary matrix?

A matrix is diagonalisable by a unitary matrix if it can be transformed into a diagonal matrix by multiplying it on both sides by a unitary matrix. This essentially means that the matrix can be simplified to a form where all its non-zero elements are on the diagonal and the rest are zeros.

3. Why is it important to prove that every unitary matrix is diagonalisable by a unitary matrix?

This proof is important because it shows that any unitary matrix can be simplified to a diagonal form, which makes it easier to work with. Diagonal matrices have many useful properties and are often used in various mathematical and scientific applications.

4. Can you provide an example of a unitary matrix that is not diagonalisable by a unitary matrix?

Yes, an example of such a matrix is the following:
[1/√2 1/√2]
[-1/√2 1/√2]
This matrix is unitary, but it cannot be diagonalised by a unitary matrix.

5. What is the proof that every unitary matrix is diagonalisable by a unitary matrix?

The proof involves using the spectral theorem, which states that any normal matrix (a matrix that commutes with its conjugate transpose) can be diagonalised by a unitary matrix. Since unitary matrices are normal, this theorem can be applied to show that every unitary matrix is diagonalisable by a unitary matrix.

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