Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

7.3 Diagonal matrices

  • Page ID
    253
  • [ "article:topic", "authorname:schilling", "Diagonalization" ]

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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 \(T\). ​