11.3: Identity Matrix

Read sections Sections 6.2 and 6.3 of the Stephen Boyd and Lieven Vandenberghe Applied Linear algebra book covers more about matrixes.

An identity matrix is a special square matrix (i.e. \(n=m\)) that has ones in the diagonal and zeros other places. For example the following is a \(3×3\) identity matrix:

\[\begin{split}
I_3 =
\left[
\begin{matrix}
1 & 0 & 0\\
0 & 1 & 0 \\
0 & 0 & 1
\end{matrix}
\right]
\end{split} \nonumber \]

We always denote the identity matrix with a capital \(I\). Often a subscript is used to denote the value of \(n\). The notations \(I_{n \times n}\) and \(I_{n}\) are both acceptable.

An identity matrix is similar to the number 1 for scalar values. I.e. multiplying a square matrix \(A_{n \times n}\) by its corresponding identity matrix \(I_{n \times n}\) results in itself \(A_{n \times n}\).

Do This

Pick a random \(3 \times 3\) matrix and multiply it by the \(3 \times 3\) Identity matrix and show you get the same answer.

Question

Consider two square matrices \(A\) and \(B\) of size \(n \times n\). \(AB=BA\) is NOT true for many \(A\) and \(B\). Describe an example where \(AB=BA\) is true? Explain why the equality works for your example.

Question

The following matrix is symmetric. What are the values for \(a\), \(b\), and \(c\)? (HINT you may want to look online or in the Boyd book for a definition of matrix symmetry)

\[\begin{split}
\left[
\begin{matrix}
3 & 5 & a\\
b & 8 & 4 \\
-3 & c & 3
\end{matrix}
\right]
\end{split} \nonumber \]

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