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Matrices 1
- An n × m matrix A is a rectangular array of numbers with n rows and m columns. By A = (aij ) we mean that aij is the entry in the ith row and the jth column. For example,
A =
[
]
is a 2 × 3 matrix. We denote by Rn×m^ the class of n × m matrices with real entries.
An n × 1 matrix is called a column vector, and a 1 × m matrix, a row vector. An n × n matrix is called square. An n × n matrix A = (aij ) is called diagonal if aij = 0 for i 6 = j. The main diagonal of A is the set of elements aii, i = 1,... , n.
The transpose of the n × m matrix A = (aij ) is the m × n matrix AT^ = (aji). Thus you get AT^ from A by transposing the rows and the columns. For example, thr tranpose of
A =
[
]
is
AT^ =
and the transpose of
x =
is xT^ = [ 1 − 2 3 ]. (2)
Note that
AT^
)T
= A.
- For reasons we’ll discuss later, we denote points in Rn^ by column vectors. For example, we write x =
[
]
instead of the (2, 1) you might be used to. To save space while observing the column vector convention, some authors will write x as [ 2 1 ]T^ or (2, 1)T^.
- Matrix Addition: If A = (aij ) and B = (bij ) are n × m matrices, then A + B is the n × m matrix with ijth entry aij + bij.
Scalar Multiplication: If A = (aij ) is an n × m matrix and c is a scalar, then cA is the n × m matrix with ijth entry caij.
We usually write −A instead of − 1 A. By A − B we mean A + (−1)B.
Matrix Multiplication: If A = (aij ) is n × m and B = (bij ) is m × k, then we can form the matrix product AB. To be precise, AB is the n × k matrix whose ijth entry is
∑m l=1 ailblj^. In other words, the ijth entry of AB is the dot product of the ith row of A with the jth column of B.
Note that matrix multiplication is not commutative. If A is n × m and B is m × k, where k 6 = n, then AB is defined, but BA is not. Even if k = n, it is not generally true that AB = BA.
Let A, B and C be matrices and k a scalar. Then,
A + (B + C) = (A + B) + C, (4)
A(B + C) = (A + B)C, (5)
(AB)C = A(BC), (6)
and k(AB) = (kA)B = A(kB), (7)
whenever the operations are defined.
- The n × n identity is the matrix I ∈ Rn×n, with 1’s one the main diagonal and 0’s elsewhere:
I =
Let I be the n × n identity. If A is n × m, then IA = A. If A is m × n, then AI = A. In particular, if A is n × n and x is n × 1, then
AI = IA = A, (9)
and Ix = x. (10)
- Vectors v 1 ,... , vm are linearly indepedent or simply independent if no one of them is a linear combination of the others. It isn’t hard to show that v 1 ,... , vm are independent if
c 1 v 1 + · · · + cmvm = 0,
implies that c 1 = c 2 = · · · = cm = 0.
In other words, the vi are linearly independent if the only linear combination of the vi that equals zero has coefficients that are all zero.
- Proposition: For j = 1,... , n, let
aj =
a 1 j .. . αnj
Let A be the n × n matrix with columns a 1 ,... , an: A = (aij ). Then det A 6 = 0 if and only if the column vectors a 1 ,... , an are linearly independent. Thus, for a square matrix A,
The columns of A are independent ⇐⇒ det A 6 = 0 ⇐⇒ A is nonsingular. (15)
- Let A be an n × m matrix. If x is in Rm, then Ax is in Rn. Thus,
A : Rm^7 → Rn.
Moreover, A(x + y) = Ax + Ay,
and A(cx) = cAx.
You can thus think of an n × m matrix A as a linear operator (or mapping, or transfor- mation) taking Rm^ to Rn. If B is k × n, then BA takes x ∈ Rm^ to Ax ∈ Rn^ and then to BAx ∈ Rk. In a nutshell, BA : Rm^7 → Rk, linearly.
You can thus think of matrix multiplication as composition of linear operators.
