MH1201 Slide Lecture 15 - Fri 14 Mar 2014 - Previous lecture Denition V nite-dimensional vector space T V V linear Then 1 T is called diagonalizable if

MH1201 Slide Lecture 15 - Fri 14 Mar 2014 - Previous...

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Previous lecture Definition V , finite-dimensional vector space T : V V , linear Then 1 T is called diagonalizable if there is an ordered basis β for V such that [ T ] β is a diagonal matrix. 2 A square matrix A is called diagonalizable if the left-multiplication L A is diagonalizable. () MH1201 3 / 3
Remark MH1201 10 / 68
Remark Suppose that D = [ T ] β is a diagonal matrix. D = [ T ] β = D 11 . . . 0 . . . 0 . . . . . . . . . . . . . . . 0 . . . D jj . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 . . . D nn . MH1201 10 / 68
Remark Suppose that D = [ T ] β is a diagonal matrix. D = [ T ] β = D 11 . . . 0 . . . 0 . . . . . . . . . . . . . . . 0 . . . D jj . . . 0 . . . . . . . . . . . . . . . 0 . . . 0 . . . D nn . Then for each vector v j β , taking into account that the j th column of D is just [ T ( v j )] β , we have T ( v j ) = D 1 j v 1 + D 2 j v 2 + · · · + D jj v j + · · · + D nj v n = D jj v j = λ j v j , where λ j = D jj . MH1201 10 / 68
MH1201 11 / 68
These observations motivate the following definitions. MH1201 12 / 68
These observations motivate the following definitions. Definition MH1201 12 / 68
These observations motivate the following definitions. Definition 1 Let T be in L ( V ) . MH1201 12 / 68
These observations motivate the following definitions. Definition 1 Let T be in L ( V ) . A non-zero vector v V is called an eigenvector of T if there exists a scalar λ such that T ( v ) = λ v . MH1201 12 / 68
These observations motivate the following definitions. Definition 1 Let T be in L ( V ) . A non-zero vector v V is called an eigenvector of T if there exists a scalar λ such that T ( v ) = λ v . The scalar λ is called the eigenvalue corresponding to the eigenvector v . MH1201 12 / 68
These observations motivate the following definitions. Definition 1 Let T be in L ( V ) . A non-zero vector v V is called an eigenvector of T if there exists a scalar λ such that T ( v ) = λ v . The scalar λ is called the eigenvalue corresponding to the eigenvector v . 2 Let A be in M n × n ( R ) . MH1201 12 / 68
These observations motivate the following definitions. Definition 1 Let T be in L ( V ) . A non-zero vector v V is called an eigenvector of T if there exists a scalar λ such that T ( v ) = λ v . The scalar λ is called the eigenvalue corresponding to the eigenvector v . 2 Let A be in M n × n ( R ) . A non-zero vector v R n is called an eigenvector of A if v is an eigenvector of L A ; that is, if Av = λ v for some scalar λ . MH1201 12 / 68
These observations motivate the following definitions. Definition 1 Let T be in L ( V ) . A non-zero vector v V is called an eigenvector of T if there exists a scalar λ such that T ( v ) = λ v . The scalar λ is called the eigenvalue corresponding to the eigenvector v . 2 Let A be in M n × n ( R ) . A non-zero vector v R n is called an eigenvector of A if v is an eigenvector of L A ; that is, if Av = λ v for some scalar λ . The scalar λ is called the eigenvalue of A corresponding to the eigenvector v . MH1201 12 / 68
Note that a vector is an eigenvector of a matrix A if and only if it is an eigenvector of L A . MH1201 13 / 68
Note that a vector is an eigenvector of a matrix A if and only if it is an

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