Solns to Axler 5C (start of)

exercises/axler-5c.md

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+# Questions from Axler 5d
+
+2, 4, 14
+
+## Question 2
+
+Suppose $A$ and $B$ are upper-triangular matrices of the same size,
+with $\alpha_1$, \dots, $\alpha_n$ on the diagonal of $A$ and
+$\beta_1$, \dots, $\beta_n$ on the diagonal of $B$.
+
+(a) Show that $A+B$ is an upper-triangular matrix with
+$\alpha_1+\beta_1$, \dots, $\alpha_n+\beta_n$ on the diagonal.
+
+(b) Show that $AB$ is an upper-triangular matrix with
+$\alpha_1\beta_1$, \dots, $\alpha_n\beta_n$. 𝛼1𝛽$ on the diagonal.
+
+### Answer
+
+(a) We must show that the leading diagonal has the form given and also
+that the entries below the leading diagonal are zero.
+
+Write $C=A+B$. We must show (i) that $C_{ii}=A_{ii}+B_{ii}$
+($i=1,\dotsc,n$); and (ii) that $C_{ij}=$ when $i>j$. Both follow
+immediately from the definition of matrix addition (3.34 in Axler)
+noting that $i>j$ implies that $A_{ij}=0$ and $B_{ij}=0$.
+
+(b) This part is slightly trickier. Write $C = AB$, so that (from
+Axler 3.41)
+
+```math
+C_{ij} = \sum_k A_{ik}B_{kj}.
+```.
+
+Start with $C_{ii}$. This term is $C_{ii}=\sum_k A_{ik}B_{ki}$. In the
+sum on the right hand side, $k$ ranges over
+$1$,\dots,$i-1$,$i$,$i+1$,\dots,$n$. But when $k<i$ the element
+$A_{ik}$ is zero; and when $k>i$, the element $B_{ki}$ is zero
+(because $A$ and $B$ are upper-triangular). So the only term which
+contrinutes to $C_{ii}$ is $A_{ii}B_{ii}$, as required.
+
+Likewise, suppose $i>j$ and consider $C_{ij}=\sum_k A_{ik}B_{kj}$. At
+least one of the multiplicands is zero when $k<i$ or when $k>j$. So
+the only terms which could be non zero are when $k$ is between $i$ and
+$j$. But $i>j$, so there are no such terms. Thus the sum is zero.
+
+
+## Question 4
+
+Give an example of an operator whose matrix with respect to some basis
+contains only 0’s on the diagonal, but the operator is invertible.
+
+(Note that the matrix is not required to be im upper-triangular form,
+only that the diagonal elements are zero.)
+
+### Answer
+