# Concept of controlled U gate in Codebook chapter I.12

Hi everyone,
First of all, I really want to thank pennylane for making such a great course material.
I am not able to understand the below lines in Codebook Chapter I.12, but I know this looks like a very very important material to develop good understanding of quantum computing.
Please explain me this concept with some basic and easy to understand easy example.

Hi @Manu_Chaudhary , thanks for checking in. Which part are you struggling with?

Thank you @Ivana_at_Xanadu for the response. For ex, in equation (6), in the matrix, there is a identity, zero matrix, U matrix. It looks quite symmetrical and but I am not getting this logic.
Is there any video to understand this? These are basic fundamentals but are very very important for developing the understanding of the subject.

Hey @Manu_Chaudhary , okay, let’s take a look. In equation (6) you’re looking at |0⟩⟨0|\otimes \mathbb{I} + |1⟩⟨1|\otimes U on the right side.
As you said, U is the unitary operator matrix from this node. \mathbb{I} is the identity, but |0⟩⟨0| is not a zero matrix!
If you look at some of the more introductory content, you will learn that |0⟩ is a non-zero vector! In fact, |0⟩ and |1⟩ are basis states for the Pauli-Z basis. In this case, |0⟩⟨0| = \left( \begin{array}{} 1 \\ 0 \end{array} \right) \left( \begin{array}{} 1 & 0 \end{array} \right) = \left( \begin{array}{} 1 & 0 \\ 0 & 0 \end{array} \right).
You can try working out the |1⟩⟨1| matrix for yourself.
Then you simply apply the Kroenecker product (\otimes) and you get the matrix on the left.

I would like to strongly encourage you to look at some of the more basic content before trying to go on with learning. For example, the first node of the introduction chapter should be able to explain most of the mathematics you got stuck on here: node I.1.

Good luck!