第四章线性代数.docVIP

4 Similar Matrices Matrix that has the simplest form is a diagonal matrix. In practice, we need to reduce a square matrix to the diagonal matrix. Thus we will introduce the definitions of eigenvalue and eigenvector，we further discuss the diagonalization of matrix. 1 Eigenvalues and Eigenvectors of Square Matrices 1、Definitions of Eigenvalue and Eigenvector DEFINITION1 Letbe an matrix，the matrix containing （1） is called the characteristic matrix of； eterminant is called the characteristic polynomial of，it is a monic polynomial of degree in ： ； （2） And equation （3） is called characteristic equation of , and the root of the characteristic equation is called the eigenvalue of. Letbe an eigenvalue of，then. Constructing the following homogeneous system ， （4） then any nonzero solution to system（4）is called an eigenvector of corresponding to the eigenvalue . EXAMPLE1 Calculate the eigenvalues and eigenvectors of. Solution The characteristic polynomial ofis ， thus the eigenvalues ofare. For，solving or . we obtain a fundamental system of solutions ，thus the eigenvectors corresponding to are ，； For，solving or . we obtain a fundamental system of solutions ，thus the eigenvectors corresponding to are ，. EXAMPLE2 Calculate the eigenvalues and eigenvectors of . Solution The characteristic polynomial of is ， thus the eigenvalues of are . For ，solving . we obtain a fundamental system of solutions ，thus the eigenvectors corresponding to are ，； For，solving . we obtain a fundamental system of solutions ，thus the eigenvectors corresponding to are ，. EXAMPLE3 Calculate the eigenvalues and eigenvectors of. Solution The characteristic polynomial of is ， thus the eigenvalues of are . For，solving . we obtain a fundamental system of solutions ，thus the eigenvectors corresponding