Properties of Matrix Operations

Addition

• A + B = B + A (commutative)
• (A + B) + C = A + (B + C) (associative)
• 0 + A = A
• (-A) + A = 0

Multiplication

• In matrix multiplication, the number of columns in the first matrix need to be equal to the number of rows in the second
  • These properties work under the assumption that the dimensions match up
• A * (B * C) = (A * B) * C
• A * (B + C) = A * B + A * C
• (B + C) * A = B * A + C * A
• I_m * A = A * I_m = A
• The identity matrix is defined as follows: $\begin{pmatrix} 1 & ... & 0 \\ \vdots & \ddots & \vdots \\ 0 & ... & 1 \\ \end{pmatrix}$
• Important properties multiplication DOES NOT have
  • Not commutative; A * B != B * A
  • AB = AC does NOT imply B = C (cannot cancel out)
  • AB = 0 does NOT imply A = 0 or B = 0

Example Showing Matrix Multiplication

$$\text{Consider the matrices } \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \text{ and } \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$$

Scalar Multiplication

$A = \begin{pmatrix} a_{11} & ... & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & ... & a_{mn} \end{pmatrix} \\ r * A = \begin{pmatrix} ra_{11} & ... & ra_{1n} \\ \vdots & \ddots & \vdots \\ ra_{m1} & ... & ra_{mn} \end{pmatrix} = \begin{pmatrix} r & ... & 0\\ \vdots & \ddots & \vdots \\ 0 & ... & r \end{pmatrix} * A = A * \begin{pmatrix} r & ... & 0\\ \vdots & \ddots & \vdots \\ 0 & ... & r \end{pmatrix}$

Matrix Transpose

A matrix with m rows by n columns can be transposed into a matrix with n rows by m columns

Examples

$\begin{pmatrix} 1 & 0 & 1 \\ 2 & 3 & 4 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 2 \\ 0 & 3 \\ 1 & 4 \end{pmatrix} \\ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} \rightarrow \begin{pmatrix} 1 & 4 & 7 & 10 \\ 2 & 5 & 8 & 11 \\ 3 & 6 & 9 & 12 \end{pmatrix} \\ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}^T = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Properties of Transpose

• (A^T)^T = A
• (A + B)^T = B^T + A^T
• r * A^T = (rA)^T
• (A * B)^T = B^T * A^T

Square Matrices

Square Matrices have an equal number of rows and columns

• If A and B are both square, then AB and BA both exist
• Square matrices can be raised to an exponent
  • A^k = A * A * … * A (k times)

Examples

$$ \begin{pmatrix} 0 & 1
1 & 0 \end{pmatrix}^2 = \begin{pmatrix} 0 & 1
1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1
1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0
0 & 1 \end{pmatrix} \

\begin{pmatrix} 1 & 1 & 0
1 & 0 & 0
0 & 0 & 1 \end{pmatrix}^3 = \begin{pmatrix} 1 & 1 & 0
1 & 0 & 0
0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 & 0
1 & 0 & 0
0 & 0 & 1 \end{pmatrix}^2
= \begin{pmatrix} 1 & 1 & 0
1 & 0 & 0
0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 1 & 0
1 & 1 & 0
0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 2 & 0
2 & 1 & 0
0 & 0 & 1 \end{pmatrix} $$

Identity Matrix

Denoted as A⁰ and I_n

$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$$

Inverse Matrices

Denoted as A^-1

• A^-1 only exists for a square matrix
• A^-1 * A = A * A^-1 = I_n
• Only exists whenever there is a pivot in every row and column

Examples

$$ \begin{pmatrix} 1 & 1
0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 1
0 & 1 \end{pmatrix}^2 = \begin{pmatrix} 1 & 2
0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 1
0 & 1 \end{pmatrix}^3 = \begin{pmatrix} 1 & 0
0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2
0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 3
0 & 1 \end{pmatrix} \

\text{Guess: the inverse is } \begin{pmatrix} 1 & -1
0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & -1
0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1
0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0
0 & 1 \end{pmatrix}
\begin{pmatrix} 1 & 1
0 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1
0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0
0 & 1 \end{pmatrix}

$$