What does it mean if one of the entries in a matrix is the Identity Matrix?

Question

i.e. take the 2x2 matrix:
[ 0 I ]
[ 0 0 ]

If I is the RxR Identity Matrix, what does that mean?

Answer 1

There's a way to represent large matrices with elements that are actually smaller matrices.

for example, the 3x3 can be:
A=
0 1 1
0 1 1
0 0 0

or in block form:
0 I
0 0

Where the 0s are 1X2, 1x1, and 2x1 matices and the I is the 2x2 identity.

The benefit is that many larger matrices are 'sparse', ie, have lots of zeros. If you block them out it simplifies computation.

If you had:another 3x3 blocked the same way:
B=
a b
c d

Then AB =
0*a+I*c 0*b+I*d
0*a+0*c 0*b+0*d
=
c d
0 0

Answer 2

How can an entry be a matrix? In your 2 x 2 example, I would have to be a 1 x 1 idsentity marix = 1.

Unless you have a conceatenated matrix

R = [A|I/B]

where A is a N x 1 column vector, I is an N x N identity matrix and B (A|I are on top of B) is a 1 x N+1 row vector.

for example transpose(A) = 1 0 1 and B = 1 0 0 0

Then [A|I/B] =
[1 1 0 0]
[0 0 1 0]
[1 0 0 1]
[1 0 0 0]

Answer 3

Pascal has it ideal, inspite of the undeniable fact that it probally be extra uncomplicated so you might place in writing out the rows and columns and cases them out. that would additionally artwork as a results of fact the entries in the matrix bypass to i,j. quite basically play around with it for slightly.