University:
University of Michigan
Course:
MATH 417 | Matrix Algebra I
Academic year:

2020
Views:

317

Pages:

1
Author:

LGarrett

J ORDAN AND GENERALIZED J ORDAN FORM OF A MATRIX Vocabulary: Jordan block, generalized Jordan block, Jordan normal form, generalized Jordan normal form Let be an element of k. We1 define the Jordan2 block by 2 3 0 0 ··· 0 61 0 · · · 07 6 7 6 · · · 07 Jn ( ) = 6 0 1 7 6 .. .. . . . . .. 7 4. . . . .5 0 0 ··· 1 (117) Show that there is a basis for k[x]/(x )n k[x] in which multiplication by x is given by the matrix Jn ( ). A matrix is said to be in Jordan normal form if it is a block matrix whose blocks are Jordan blocks. (118) Suppose that the field k is algebraically closed. Show that each n ⇥ n matrix with entries in k is similar to a matrix in Jordan normal form, and that the Jordan normal form is unique up to reordering blocks. Let f = xd + fd 1 xd 1 + · · · + f1 + 0 be a monic polynomial with coefficients in k. Let Ud be the d ⇥ d matrix with a 1 in the upper-right corner and all other entries 0. Define the generalized Jordan block Jn (f (x)) to be the (dn) ⇥ (dn) block matrix 2 3 C(f ) 0 0 ··· 0 6 Ud C(f ) 0 ··· 0 7 6 7 6 0 Ud C(f ) · · · 0 7 Jn (f ) = 6 7 6 .. .. .. 7 .. .. 4 . . . . 5 . 0 0 · · · Ud C(f ) n (119) Show that there is a basis for k[x]/f (x) k[x] in which multiplication by x is given by the matrix Jn (f (x)). Define a matrix to be in generalized Jordan normal form3 if it is a block diagonal matrix where each block is of the form Jni (pi (x)) and the polynomials pi (x) are irreducible. (120) Show that each n ⇥ n matrix with entries in k is similar to a matrix in generalized Jordan normal form, and that the generalized Jordan normal form is unique up to reordering blocks. 1The more standard choice is to take J ( ) to be the transpose of this. The choice given here is more compatible with the standard choices used n to define rational canonical form, so we will adopt it. There is no important difference between these conventions. 2 Named for Camille Jordan, French mathematician, 1838-1922. 3Wikipedia says that some people call this principal rational canonical form.

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