Commits

rbeezer committed 8f16403

Change: Proof of Theorem OD uses normality of diagonal matrices (Hunter Wills)

Comments (0)

Files changed (3)

+v3.01 2013/xx/xx
+~~~~~~~~~~~~~~~~
+Change:  Proof of Theorem OD uses normality of diagonal matrices (Hunter Wills)
+
 v3.00 2012/12/05
 ~~~~~~~~~~~~~~~~
 New:  Diagrams NSLT, SLT (surjective linear transformations)
 
 <isbn>978-0-9844175-5-1</isbn>
 
-<version>3.00</version>
+<version>3.01-preview</version>
 
 <title>A First Course in Linear Algebra</title>
 

src/section-OD.xml

 
 <!--   TODO:  Proof needs some results, like adjoint of a product -->
 <proof>
-<p><implyforward /> Suppose there is a unitary matrix $U$ that diagonalizes $A$.  We would usally write this condition as $\adjoint{U}AU=D$, but we will find it convenient in this part of the proof to use our hypothesis in the equivalent form, $A=UD\adjoint{U}$.  Notice that since $D$ is a diagonal matrix, $\adjoint{D}$ is also a diagonal matrix, and therefore $D$ and $\adjoint{D}$ commute when multiplied.  We check the normality of $A$,
+<p><implyforward /> Suppose there is a unitary matrix $U$ that diagonalizes $A$.  We would usally write this condition as $\adjoint{U}AU=D$, but we will find it convenient in this part of the proof to use our hypothesis in the equivalent form, $A=UD\adjoint{U}$.  Recall that a diagonal matrix is normal, and notice that this observation is at the center of the next sequence of equalities.  We check the normality of $A$,
 <alignmath>
 \adjoint{A}A
 <![CDATA[&=\adjoint{\left(UD\adjoint{U}\right)}\left(UD\adjoint{U}\right)&&]]>\text{Hypothesis}\\
 <![CDATA[&=U\adjoint{D}\adjoint{U}UD\adjoint{U}&&]]>\text{<acroref type="theorem" acro="AA" />}\\
 <![CDATA[&=U\adjoint{D}I_{n}D\adjoint{U}&&]]>\text{<acroref type="definition" acro="UM" />}\\
 <![CDATA[&=U\adjoint{D}D\adjoint{U}&&]]>\text{<acroref type="theorem" acro="MMIM" />}\\
-<![CDATA[&=UD\adjoint{D}\adjoint{U}&&]]>\text{<acroref type="property" acro="CMCN" />}\\
+<![CDATA[&=UD\adjoint{D}\adjoint{U}&&]]>\text{<acroref type="definition" acro="NRML" />}\\
 <![CDATA[&=UDI_{n}\adjoint{D}\adjoint{U}&&]]>\text{<acroref type="theorem" acro="MMIM" />}\\
 <![CDATA[&=UD\adjoint{U}U\adjoint{D}\adjoint{U}&&]]>\text{<acroref type="definition" acro="UM" />}\\
 <![CDATA[&=UD\adjoint{U}\adjoint{\left(\adjoint{U}\right)}\adjoint{D}\adjoint{U}&&]]>\text{<acroref type="theorem" acro="AA" />}\\