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Find unitary matrix $U\in\mathbb{C}^{2\times2}$ so that $D=UAU^*$ is diagonal where $$A=\begin{bmatrix} 3 & -4\\4 &3 \end{bmatrix} \in \mathbb{R}^{2\times 2}$$

I know that for unitary matrix it holds that $U^* U = U U^* = I$ and if we know that $D$ is diagonal:

\begin{align} D&= U^* A U \;\;\;|\cdot U\\ UDU^*&=A \;\;\;\;\;\;\;\;\;\;\,| \cdot U^* \end{align}

Do I start to find the eigenvalues & vectors for $A$? So if $A$ is diagonalizable then for it holds $A=PDP^{-1}$ and because $U$ is unitary $P=U$ in this case? So first the characteristic polynomial:

\begin{align} f(\lambda_A) &= \operatorname{det}(A-I\lambda)=\lambda ^2-6 \lambda +25\\ \text{Roots are} \\ f(\lambda_A) &=0\to\lambda=3-4 i\lor \lambda =3+4 i \end{align} Corresponding eigenvectors \begin{align} Ax&=\lambda x \\ \begin{bmatrix} 3 & -4\\4 &3 \end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix} &=\begin{bmatrix} 3x_1-4ix_1\\ 3x_2-4ix_2\end{bmatrix} \\ 3x_1-4x_2 &= 3x_1-4ix_1 \\ x_2 &=ix_1\to x=\begin{bmatrix}1 \\ i \end{bmatrix} \end{align} For $\lambda=3+4i \to x=\begin{bmatrix}1 \\ -i\end{bmatrix}$. (Mathematica gave me $\left( \begin{array}{cc} i & 1 \\ -i & 1 \\ \end{array} \right)$ for some reason...) So according to decomposition $A=PDP^{-1}$ it holds that $D=\begin{bmatrix}3-4i & 0 \\ 0 & 3+4i\end{bmatrix}$ and $P=\begin{bmatrix}1 & 1 \\ i & -i \end{bmatrix}= U$? Is this correct or did I fumble somewhere?

Test:

\begin{align} \begin{bmatrix}1 & 1 \\ i & -i \end{bmatrix}\begin{bmatrix}3-4i & 0 \\ 0 & 3+4i\end{bmatrix}\begin{bmatrix}1 & 1 \\ i & -i \end{bmatrix}^*&=A \\ \begin{bmatrix}3-4i & 3+4i \\ 4+3i & -3-4i\end{bmatrix}\begin{bmatrix}1 & -i\\ 1 & -i\end{bmatrix}&=\begin{bmatrix} 6 & -8 \\ 8 & 6 \end{bmatrix} \neq A \end{align}

So I fumbled. Hints?

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  • $\begingroup$ You can just multiply the matrices to check for yourself. $\endgroup$ – Marc van Leeuwen Dec 3 '15 at 14:46
  • $\begingroup$ Okey so I screwed up somewhere. $\endgroup$ – ELEC Dec 3 '15 at 15:03
  • $\begingroup$ You're on the right track. Your eigenvalues and eigenvectors look fine. However, your $P$ isn't quite right: The $-1$ should be $-i$, shouldnt it? Additionally: Although the columns of $P$ are orthogonal, do we have that $P^* P = I $ and $P^2 = P$? Should $U$ contain the eigenvectors as columns or rows? $\endgroup$ – Roland Dec 3 '15 at 15:28
  • $\begingroup$ $PP^* = \left( \begin{array}{cc} 2 & 0 \\ 0 & 2 \\ \end{array} \right)$. So for $P = P D P^{-1}$ to hold $D=\frac 12 I \lambda$?. $\endgroup$ – ELEC Dec 3 '15 at 15:55
  • $\begingroup$ @Roland Whats the reasoning that my results is $2 A$ ? How do I account for it. I'm not sure about the "Should $U$ contain the eigenvectors as columns or rows?" either $\endgroup$ – ELEC Dec 3 '15 at 16:09
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As we're looking for a unitary matrix $U$ which diagonalizes $A= \begin{pmatrix} 3& -4\\4&3\end{pmatrix},$ such that $U^*AU =D$ it's a good idea to look for the eigenvalues - we know that $P^{-1}A P = D$ is a diagonal matrix which contains the eigenvalues when $P$ is an invertible matrix containing the eigenvectors as columns. This looks similar to the required equation, but not quite.

An orthogonal matrix satisfies $U^* U =U U^*=I$. If we denote by $u_1, \dots, u_n$ the columns of $U$, this can also be rephrased as

$$(u_i,u_i)= \overline u_i ^T u_i = 1$$ and $$(u_i,u_j)= \overline u_i ^T u_j = 0,$$

i.e. the column vectors of $U$ have norm $1$ and are orthogonal to each other. If this is the case, then $U^* = U^{-1}.$

In order to diagonalize $A$ with an orthogonal matrix, we need not any matrix $P$ which diagonalizes, but an orthogonal matrix.

Calculate the eigenvalues of $A$ as above: $\lambda_\pm = 3 \pm 4i$.

Solve $(A - \lambda_\pm )v_\pm = \begin{pmatrix} \mp 4i& -4\\4&\mp 4i\end{pmatrix} v_\pm=0$ and get some eigenvectors $v_+ = \begin{pmatrix} 1 \\i \end{pmatrix}, v_- = \begin{pmatrix} 1 \\-i \end{pmatrix}$.

If we put these as columns of a matrix, will the matrix be orthogonal? We'll check with the scalar product which has a complex conjugate for complex vector spaces (the lack of this might be the reason you end up with a norm of 0).

$(v_+,v_-)= \overline{\begin{pmatrix} 1 & i \end{pmatrix}}\begin{pmatrix} 1 \\-i \end{pmatrix}= 1*1 + (-i)*(-i)=1-1=0,$ which shows that $v_+$ and $v_-$ are indeed orthogonal.

But what's the norm of these vectors? We have $(v_+,v_+)= \overline{\begin{pmatrix} 1 & i \end{pmatrix}}\begin{pmatrix} 1 \\i \end{pmatrix}= 1*1 + (-i)*i= 1 + 1 = 2$ and analogously $(v_-,v_-)=2$, so both vectors don't have norm $1$, but norm $\sqrt{2}$. We can fix this by taking $u_1 = \frac{1}{\sqrt{2}}v_+, \ u_2 = \frac{1}{\sqrt{2}}v_-$ and defining $U$ as the matrix which contains these as columns.

This is a special choice for $P$ which will diagonalize $A$, but which also is orthogonal so that we can use $P^*$ instead of $P^{-1}$.

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  • $\begingroup$ Got it! :) very nice $\endgroup$ – ELEC Dec 3 '15 at 19:22
  • $\begingroup$ Almost. Please check your definition of the norm you're using (my guess is that you've forgottent the complex conjugation). The only vector which has norm zero is the zero vector. Orthogonality refers to each other, not to each vector itself. $\endgroup$ – Roland Dec 3 '15 at 19:22
  • $\begingroup$ Oh! Didn't know about complex norm $||\mathbf{x_1}||=\sqrt{|1/\sqrt{2}|^2 + |i/\sqrt{2}|^2} = \sqrt{0.5+0.5}=1$. $\endgroup$ – ELEC Dec 3 '15 at 19:25
  • $\begingroup$ Yeah and now they are orthogonal as the eigenvectors innerproduct is zero right? $\endgroup$ – ELEC Dec 3 '15 at 19:27
  • $\begingroup$ $\|x_1\| = \sqrt{\overline{\frac{1}{\sqrt{2}}}*\frac{1}{\sqrt{2}} + \overline{\frac{i}{\sqrt{2}}}*\frac{i}{\sqrt{2}}} = \sqrt{0.5 + 0.5}$ $\endgroup$ – Roland Dec 3 '15 at 19:28

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