MATRIX OPERATIONS

Matrix Addition and Scalar Multiply:

for $m \times n$ A, B and C

• A+B is defined by (A+B)_ij = a_ij + b_ij.
• rA for scalar r is defined by (rA)_ij = ra_ij.
• Properties

  1.

  A+B=B+A

  2.

  (A+B)+C=A+(B+C)

  3.

  A+O=A, zero matrix, additive identity

  4.

  r(A+B)=rA+rB, scalar r

  5.

  (r+s)A=rA+sA, scalars r and s

  6.

  r(sA)=(rs)A, scalars r and s

Matrix Multiply:

for an $m \times n$ A, $n \times p$ B and $p\times k$ C

• Definition
  $AB = A[{\bf b}_1\ {\bf b}_2\ \ldots\ {\bf b}_p]= [A{\bf b}_1\ A{\bf b}_2\ \ldots\ A{\bf b}_p]$, or
  $(AB)_{ij} = a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{in}b_{nj}$; needs mnp scalar multiplications.
• Properties

  1.

  A(BC) = (AB)C

  2.

  A(B+C) = AB+AC

  3.

  (B+C)A = BA+CA

  4.

  r(AB) = (rA)B = A(rB) = (AB)r, scalar r

  5.

  I_mA = AI_n, identity multiplication

Commutativity:

If AB = BA, then A and B are said to commute.
NOTE: $AB \neq BA$ USUALLY!.

Powers:

A^k = A^k-1A; A⁰ = I, A¹ = A, A² = AA, A³ = A²A = AAA, etc.
Note: Compute $A^k{\bf x}$ using $A(A^{k-1}{\bf x})$ not $(A^k){\bf x}$.

Transpose:

(A^T)_ij = (A)_ji; (AB)^T = B^TA^T.

Scalar, Inner or Dot Product:

${\bf x}^T{\bf y}= x_1y_1+\cdots +x_ny_n$(sometimes denoted by ${\bf x}\cdot {\bf y}$)
Note: if ${\bf u}_i^T$ is row i or A, $(AB)_{ij} = {\bf u}_i^T{\bf b}_j$.

Outer Product:

${\bf x}{\bf y}^T$ - an ($n\times 1$) ${\bf x}$ times a ($1\times n$) ${\bf y}$ is an $n\times n$ matrix,

• 
  
  $${\bf x}{\bf y}^T = \left[ \begin{array}{cccc} x_1y_1 & x_1y_2... ...y_n \end{array}\right];\ \ \ ({\bf x}{\bf y}^T)_{ij} = x_iy_j.$$
  
  

• Outer product form for AB: if ${\bf v}_i^T$ is row i or B, then $AB = {\bf a}_1{\bf v}_1^T + {\bf a}_2{\bf v}_2^T + \cdots + {\bf a}_n{\bf v}_n^T$.

MATRIX INVERSES

Inverse of A:

For an $n\times n$ A

• Definition: if, for some $n\times n$ C, AC = CA = I, then A is invertible, and C is the inverse of A, notated by A^-1.
• $A{\bf x}= {\bf b}$? If A is invertible, then $A^{-1}A{\bf x}= A^{-1}{\bf b}$, so ${\bf x}= A^{-1}{\bf b}$.
• Properties: (A^-1)^-1 = A, (AB)^-1 = B^-1A^-1, (A^T)^-1=(A^-1)^T.

Finding A^-1:

• n = 2
  
  $$\mbox{If}\ A = \left[\begin{array}{cc}a & b \\ c & d \end{arr... ...bc}\left[\begin{array}{cc}d & -b \\ -c & a \end{array}\right].$$
  
  

• General n
  
  • Use $[A \ \ I] \sim [I \ \ A^{-1}]$, or
  • solve $A{\bf x}_i = \be_i$, for $i = 1, 2, \ldots, n$; then $A^{-1}= X = [{\bf x}_1\ {\bf x}_2\ \ldots\ {\bf x}_n]$, or
  • use inverse formula (inefficient for n > 3, discussed in later lesson).