7.3: Diagonal matrices
- Page ID
- 253
Note that if \(T\) has \(n=\dim(V)\) distinct eigenvalues, then there exists a basis \((v_1,\ldots,v_n)\) of \(V\)such that
\begin{equation*} Tv_j = \lambda_j v_j, \quad \text{for all \(j=1,2,\ldots,n\).}
\end{equation*}
Then any \(v\in V\) can be written as a linear combination \(v=a_1v_1+\cdots+a_nv_n\)of \(v_1,\ldots,v_n\). Applying \(T\) to this, we obtain \begin{equation*}
Tv = \lambda_1 a_1 v_1 + \cdots + \lambda_n a_n v_n.
\end{equation*}
Hence the vector \begin{equation*}
M(v) = \begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix}
\end{equation*}
is mapped to \begin{equation*}
M(Tv) = \begin{bmatrix} \lambda_1 a_1 \\ \vdots \\ \lambda_n a_n \end{bmatrix}.
\end{equation*}
This means that the matrix \(M(T)\)for \(T\) with respect to the basis of eigenvectors \((v_1,\ldots,v_n)\) is diagonal, and so we call \(T\) diagonalizable: \begin{equation*}
M(T) = \begin{bmatrix} \lambda_1 & & 0 \\ & \ddots & \\
0& & \lambda_n \end{bmatrix}.
\end{equation*}
We summarize the results of the above discussion in the following Proposition.
Proposition 7.3.1. If \(T\in \mathcal{L}(V,V)\) has \(\dim(V)\) distinct eigenvalues, then \(M(T)\) is diagonal with respect to some basis of \(V\). Moreover, \(V\) has a basis consisting of eigenvectors of \(T\).
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
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