Linear Algebra Notes
Thomas Goller
September 4, 2011
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Rational and Jordan Canonical Forms
For the rational form (over any field), the cyclic decomposition theorem guarantees the existence of non-zero vectors α1 , . . . , αr with respective T -annihilators
p1 , . . . , pr such that
V = Z(α1 ; T ) ⊕ · · · ⊕ Z(αr ; T )
and
pk |pk−1 ,
k = 2, . . . , r,
with p1 = p the minimal polynomial of T , p1 · · · pr = f the characteristic
polynomial, and deg pi = dim Z(αi ; T ). Then in the basis
{α1 , T α1 , . . . , T deg p1 −1 α1 , α2 , T α2 , . . . },
T is composed of blocks

0 0 0
1 0 0

0 1 0

 .. .. ..
. . .
0 0 0
···
···
···
0
0
0
..
.
−a0
−a1
−a2
..
.
···
1
−ak−1




,


where pi = xk + ak−1 xk−1 + · · · + a0 .
The Jordan form (over an algebraically closed field like C) is obtained by
combining the primary decomposition theorem with the rational form, as follows. Given A with characteristic polynomial f = (x − c1 )d1 · · · (x − ck )dk and
minimal polynomial p = (x − c1 )r1 · · · (x − ck )rk , with 1 ≤ ri ≤ di for each i, we
use the primary decomposition theorem to write
V = W1 ⊕ · · · ⊕ Wk ,
where Wi is the null space of (T − ci I)ri , and Wi has dimension di . We choose a
basis for Wi corresponding to the cyclic decomposition of the nilpotent operator
1
Ni := T − ci I on Wi . Since Ni has minimal polynomial xri and T = Ni + ci I,
the matrix for T in the chosen basis is composed of blocks of the form


ci 0 · · · 0 0
 1 ci · · · 0 0 


 .. ..
.. ..  ,
. .
. .




ci
0 0 · · · 1 ci
decreasing in size, with the first of size ri and the sum of the sizes equal to di .
Putting these result together for all the Wi gives the Jordan form.
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Special Matrices
Let V be a finite-dimensional inner product space.
• For every self-adjoint linear operator T on V , there is an orthonormal
basis of V consisting of characteristic vectors of T .
• It follows that Hermitian matrices are diagonalizable by unitary matrices, and real symmetric matrices are diagonalizable by real orthogonal
matrices.
• Even stronger: if T is a normal operator on V , there is an orthonormal
basis of V consisting of characteristic vectors of T . Hence every normal
matrix is diagonalizable by a unitary matrix.
2