Definition 1.11.1.
Square matrices \(A,B\in M_n(\K)\) are said to be similar if there exists an invertible matrix \(P\in M_n(\K)\) such that \(A=PBP^{-1}\text{.}\)
Section 1.11 Diagonalization of a square matrix
We introduce diagonalization of square matrices of small sizes in this section.
Note 1.11.2.
If \(A\) similar to \(B\) then \(B\) is also similar to \(A\text{.}\)
Definition 1.11.3.
A square matrix \(A\in M_n(\K)\) is said to be diagonalizable if \(A\) similar to a diagonal matrix in \(\K\text{.}\)
Definition 1.11.4.
Consider a nonzero polynomial \(p(x)\in\K[x]\text{.}\) Suppose that \(p(x)\) can be factored in \(\K[x]\) as follows.
\begin{equation*}
p(x)=a(x-\lambda_1)^{n_1}(x-\lambda_2)^{n_2}\cdots(x-\lambda_r)^{n_r}
\end{equation*}
where, \(a\in\K\) is some nonzero element, \(n_i\) are some natural numbers, and \(\lambda_i\) are distinct elements of \(\K\text{,}\) i.e., \(\lambda_i\neq\lambda_j\) for \(i\neq j\text{.}\)
We say that the multiplicity of \(\lambda_i\) is \(n_i\text{.}\)
Definition 1.11.5.
Suppose that \(A\in M_n(\K)\) is a square matrix. Let \(\ell_A\) be the linear map associated to \(A\text{,}\) i.e., \(\ell_A\colon M_{n\times 1}(\K)\to M_{n\times 1}(\K)\) is given by \(\ell_A(v)=Av\text{.}\) We define the kernel of \(A\) to be the same as the kernel of the linear map \(\ell_A\text{.}\)
We denote the kernel of \(A\) by \(\ker(A)\text{.}\)
Definition 1.11.6.
Suppose that \(A\in M_n(\K)\) is a square matrix. We define the dimension of the kernel of \(A\) to be \(n-{\rm rank}(A)\text{.}\)
We denote the dimension of the kernel of \(A\) by \(\dim(\ker(A))\). Thus we have the following.
\begin{align*}
\dim\left(\ker(A)\right) \amp=\text{ size of the matrix }A-\text{ rank of the matrix }A \\
\amp=n-{\rm rank}(A)
\end{align*}
Remark 1.11.7.
The dimension of the kernel is defined in linear algebra using the concept of linear independence. The Definition 1.11.6 is in fact ‘Rank-Nullity Theorem’. Due to lack of time we take Definition 1.11.6 as a working definition.
Fact 1.11.8.
Let \(A\in M_n(\K)\) be a square matrix. Suppose that the characteristic polynomial of \(A\) has the following factorization with \(\lambda_i\neq\lambda_j\) for \(i\neq j\text{.}\)
\begin{equation*}
\chi_A(x)=(x-\lambda_1)^{n_1}(x-\lambda_2)^{n_2}\cdots(x-\lambda_r)^{n_r}\in\K[x]
\end{equation*}
The matrix \(A\) is diagonalizable if and only if for every \(1\leq i\leq r\) we have the following.
\begin{equation*}
\dim\left(\ker(\lambda_iI_n-A)\right)=n_i
\end{equation*}
If \(A\) is diagonalizable then it is similar to the following diagonal matrix.
\begin{equation*}
\begin{pmatrix}\lambda_1I_{n_1}\amp\amp\amp\amp\\\amp\lambda_2I_{n_2}\amp\amp\amp\amp\\\amp\amp\ddots\amp\amp\amp\\\amp\amp\amp\amp\lambda_rI_{n_r}\end{pmatrix}
\end{equation*}
where, each \(\lambda_iI_{n_i}\) is \(n_i\times n_i\) diagonal matrix with all diagonal entries \(\lambda_i\) and all other entries \(0\text{.}\) Furthermore, in the above matrix entries left blank are taken to be \(0\text{.}\)
