Linear Algebra (Math 220)
Assignment due Tuesday, April 29

The Characteristic Equation

If, relative to a given coordinate system in an $n$-dimensional vector space $V$, the columns of an invertible $n\times n$ matrix $Q$ form a basis of that vector space relative to which a linear transformation that is represented in the given coordinate system by a matrix $M$ is diagonalized, i.e., represented by a diagonal matrix $D$, then $$Q^{-1}MQ=D\text{.}$$ Equivalently $MQ=QD$, and, taking the $j^{\mathrm{th}}$ column one sees that $$M{Q}_j={\left(MQ\right)}_j={\left(QD\right)}_j=Q{D}_j=d_{jj}{Q}_j\text{.}$$ Thus, the member $Q_j$ of the diagonalizing basis must lie in the kernel of the linear function represented in the given coordinate system by the matrix $M-d_{jj}{1}_n$, where $1_n$ denotes the $n\times n$ identity matrix. Thus, each $Q_j$ may be found by computing the kernel of $M-t{1}_n$ when $t=d_{jj}$, and the diagonal elements $d_{jj}$ of $D$ may be found among the roots of the characteristic polynomial of $M$ $$\det \left(M-t{1}_n\right)=0\text{.}$$ A root of the characteristic polynomial is called an eigenvalue of $M$ and a coordinate column $v\ne 0$ with the property that $Mv=\lambda v$ for some eigenvalue $\lambda$ of $M$ is called an eigenvector of $M$. (Moreover, the eigenvalues of $M$ and the elements of $V$ represented in the given coordinate system by the eigenvectors of $M$ may also be called eigenvalues and eigenvectors of the underlying linear transformation of $V$.)