Matrices, determinants

A square matrix (for which m = n) is called a diagonal matrix if all
elements aij for which i 6= j are zero.
If all elements aii are one, then the matrix is called an identity
matrix, denoted with Im (depending on the context, the subscript
m may be left out).
I =
0
BBB@
1 0


0
0 1


0
...
...
. . .
...
0 0


1
1
CCCA
If all matrix entries are zero (i.e. aij = 0 for all i; j), then the
matrix is called a zero matrix, denoted with 0.
Similarly, we can dene subtraction between an mA  nA matrix A
and an mB  nB matrix B as
A ?? B = C, with cij = aij ?? bij
for all 1  i  mA;mB and 1  j  nA; nB.
For example:
0
@
1 4
2 5
3 6
1
A ??
0
@
9 12
8 11
7 10
1
A =
0
@
??8 ??8
??6 ??6
??4 ??4
1
A
Again, for the dimensions of the matrices A and B we must have
mA = mB and nA = nB.
Graphics 2011/2012, 4th
Matrix multiplication is distributive over addition:
A(B + C) = AB + AC
(A + B)C = AC + BC
and it is associative:
(AB)C = A(BC)
However, it is not commutative, i.e. in general,
AB is not the same as BA.
GrIdentity matrix Im:
I =
0
BBB@ 1
0






0
0 1


0
...
...
. . .
...
0 0


1
1
CCCA
With matrix multiplication we get IA = AI = A
(hence the name \identity matrix").
G