Table of Contents - 11.4.4. Ring isomorphism between polynomial matrices and polynomials over matrices

The main result of this section is theorem pmmpric, which shows that the
ring of polynomial matrices and the ring of polynomials having matrices as
coefficients (called "polynomials over matrices" in the following) are
isomorphic:<br>
<br><br>
Or in a more common notation:<br>
corresponds to
M(n, R[t]), the ring of n x n polynomial matrices over the ring R.<br>
corresponds to M(n, R)[t], the
polynomial ring over the ring of n x n matrices with entries in ring R.
<br><br>
<br><br>
with and ( i m j ) ) is an isomorphism
between these rings:<br>
<br>
with (see pm2mpf1o and pm2mprngiso), and<br>
<br>
p )
<br><br>
is the corresponding inverse function:<br>
<br>
(see mp2pm2mp).
<br><br>
In this section, the following conventions are mostly used:
<ul>
<li> is a (unital) ring (see df-ring)</li>
<li> is the polynomial algebra over (the ring) (see
df-ply1)
<ul>
<li> is its base set (see df-base)</li>
<li> is its variable (see df-vr1)</li>
<li> is its scalar multiplication (see df-vsca or
lmodvscl)</li>
<li> is its exponentiation (see df-mulg)
</li>
</ul></li>
<li> is the algebra of N x N matrices over (the ring)
(see df-mat)</li>
<li> is the algebra of N x N matrices over (the polynomial
ring) .
<ul>
<li> is its base set</li>
<li> is a concrete polynomial matrix</li>
</ul></li>
<li> is the polynomial algebra over (the matrix ring)
.
<ul>
<li> is its base set</li>
<li> is a concrete polynomial with matrix coefficients</li>
<li> is its variable</li>
<li> is its scalar multiplication</li>
<li> is its exponentiation</li></ul></li>
</ul>