Question

What is the result if I "DIRECT SUM" the following things:
Matrix +Direct Sum+ Constant Number
Thanks.

Answer 1

if \(\textbf M=\textbf M_{ab}=\left(\begin{array} \ m_{11}& \dots&m_{1b} \\ \vdots&\ddots &\vdots\\ m_{a1}&\dots& m_{ab}\end{array} \right)\)

\[\textbf M\oplus c=\left(\begin{array} \ m_{11}& \dots&m_{1b} &0\\ \vdots&\ddots &\vdots&0\\ m_{a1}&\dots& m_{ab}&0\\ 0 &0&0&c\end{array} \right)\]

Answer 2

thank you !!
now, if you have time, do you mind explaining the theory behind why this is the case?

Answer 3

the direct sum just when you combine the matrices along the main diagonal ,
in this case the second matrix is just a scalar

Answer 4

if \(\textbf M_{ab}=\left(\begin{array} \ m_{11}& \dots&m_{1b} \\ \vdots&\ddots &\vdots\\ m_{a1}&\dots& m_{ab}\end{array} \right)\)and \(\textbf N_{cd}=\left(\begin{array} \ n_{11}& \dots&n_{1d} \\ \vdots&\ddots &\vdots\\ n_{c1}&\dots& n_{cd}\end{array} \right)\)



\[\textbf M_{ab}\oplus \textbf N_{cd}=\left(\begin{array} \ m_{11}& \dots&m_{1b} &0&0&0\\ \vdots&\ddots &\vdots&0&0&0\\ m_{a1}&\dots& m_{ab}&0&0&0\\ 0 &0&0& \ n_{11}& \dots&n_{1d} \\ 0&0&0& \vdots&\ddots &\vdots\\0 &0&0& n_{c1}&\dots& n_{cd}\end{array} \right)\]

Answer 5

see?

Answer 6

yeh i see ! i just have to understand why its down the diagonal :) !
thanks for your help, i appreciate it greatly !

Answer 7

thats just part of the definition of direct sum

Answer 8

i see ! thank again unkle :)

Answer 9

the rows and columns already have a variables associated with them, adding more information to a matrix requires more variables, hence more rows and columns

Answer 10

i see !

Answer 11

also, dim(N) +dim (M) = dim(N+M), in correspondance to direct sums, which means there has to be a bigger matrix

Answer 12

makes sense