MATH136 Lecture Notes - Lecture 13: Identity Matrix, Main Diagonal, Transpose
14 May 2018
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Monday, May 29
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Lecture 13 : Matrix multiplication. (Refers to section 3.1)
Concepts:
1. matrix multiplication.
2. multiplying two matrices.
3. block-multiplying two matrices.
13.1 Definition of “matrix multiplication” between two matrices A and B of dimension
m × n and n × k, respectively.
Operation What we start with : What we get :
Matrix
multiplication
13.1.1 Examples
a) Compute the (2, 3)th entry of the product DE.
Answer: 0
b) Compute the (2, 3)th entry of the product ETDT for the matrices D and E above.
Answer: Let B = [bij ] = ETDT. Then b23 = r2 ⋅ c3 = cE2 ⋅ rD3 = (1, 4) ⋅ (1, −7) = −27
13.1.2 Definition − Let An × n = [aij] be a square matrix. The entries {a11, a22, a33, …
ann} is referred to as being the main diagonal of A. The matrix A is called the
identity matrix of dimension n, if every element on the main diagonal of A is 1 while
all others are zero. That is, aii = 1 for all i = 1 to n and aij = 0 whenever i ≠ j. The
identity matrix is denoted by I or In × n.
Note – Two matrices Am × n and Bp × k are said to be multipliable only if n snd p are
equal.
13.1.3 Example − Consider the matrix A and the two following identity matrices:
We see that I plays the role of the “multiplicative identity” in matrix algebra.
13.2 Rules of matrix algebra − We previously listed a few addition, scalar multiplication
and transposition properties in lecture 12. We now add a few more matrix algebra
properties involving matrix-multiplication. All of these can be proven directly from the
definition. Most proofs are not difficult but rather tedious to write out. In this list we
assume that pairs of matrices which are multiplied are indeed multipliable.
a) (αβ)A = α(βA)
b) A(BC) = (AB)C
c) In × n An × m =A, An × mIm × m = A
d) (A + B)C= AC + BC
e) C (A + B) = CA + CB
f) (αA)B= A(αB)
g) (A B)T = BTAT
h) (αA)T= αA T
13.3 Observation − Looking over the rules of matrix algebra above we see that operations
(addition, scalar multiplication and multiplication) on matrices are very similar to
operations on numbers and/or vectors. They “behave” in the way we expect them to and
appear natural. There are however two significant rules that do differ from what we might
expect:
1. Multiplication of matrices is not commutative; i.e., A m × n B n × m ≠ BA. (Since A and B
might be multipliable as AB but not in as BA since their dimensions don’t allow it).
Even if both B and A are square matrices with the same dimension n × n it may still
happen that AB ≠ BA.
2. Transpose of a product: Notice this interesting property: (AB)T = BTAT.
It is not true in general that (AB)T = ATBT.
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Document Summary
Monday, may 29 lecture 13 : matrix multiplication. (refers to section 3. 1) Concepts: matrix multiplication, multiplying two matrices, block-multiplying two matrices. 13. 1 definition of matrix multiplication between two matrices a and b of dimension m n and n k, respectively. 13. 1. 1 examples: compute the (2, 3)th entry of the product de. Answer: 0: compute the (2, 3)th entry of the product etdt for the matrices d and e above. Answer: let b = [bij ] = etdt. Then b23 = r2 c3 = ce2 rd3 = (1, 4) (1, 7) = 27. 13. 1. 2 definition let an n = [aij] be a square matrix. The entries {a11, a22, a33, ann} is referred to as being the main diagonal of a. The matrix a is called the identity matrix of dimension n, if every element on the main diagonal of a is 1 while all others are zero.
