MATH 217 Chapter Notes - Chapter 7: Diagonalizable Matrix, Invertible Matrix, Diagonal Matrix
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Consider a linear transformaion t(x) = ax from rn to rn. A is said to be diagonalizable if the matrix b of t with respect to some basis is diagonal. Matrix a is diagonalizable if and only if a is similar to some diagonal matrix b, meaning that there exists an inverible matrix s such that s^-1as=b is diagonal. A nonzero vector v in rn is called an eigenvector of a if av = v for some scalar . If the vectors v1 vn are eigenvectors of a, meaning that av1= 1v1 . avn = nvn for some scalars 1 n, then v1 . vn is an eigenbasis. The sum of the diagonal entries of a square matrix a is called the trace of a, denoted by tr(a) The kernel of the matrix a- in is called the eigenspace associated with , denoted e : Consider an eigenvalue of an nxn matrix a.
