Matrix Structure

Let \(A\) be an \(m \times n\) matrix:

\[ A = \begin{bmatrix} \vec{a}_1 & \vec{a}_2 & \dots & \vec{a}_n \end{bmatrix} \]

We can write the entries of \(A\) explicitly as follows:

\[ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1j} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2j} & \dots & a_{2n} \\ \vdots & \vdots & & \vdots & & \vdots \\ a_{i1} & a_{i2} & \dots & a_{ij} & \dots & a_{in} \\ \vdots & \vdots & & \vdots & & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mj} & \dots & a_{mn} \end{bmatrix} = [a_{ij}]_{m \times n} \]

Definitions

Diagonal Matrix: A square \(n \times n\) matrix whose non-diagonal entries are zero. For example:

\[ \begin{bmatrix} a_{11} & 0 & \dots & 0 \\ 0 & a_{22} & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & a_{nn} \end{bmatrix} \]

Zero Matrix: A matrix whose entries are all zero (denoted by \(O\)).

\(n \times n\) Identity Matrix:

\[ I_n = \begin{bmatrix} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 \end{bmatrix} \]

(I) Sums and Scalar Multiples

Two matrices are equal (\(A = B\)) if they have the same size and the same corresponding entries.

Sum: Let \(A = [a_{ij}]_{m \times n}\) and \(B = [b_{ij}]_{m \times n}\). Their sum is defined as:

\[ A + B = [a_{ij} + b_{ij}]_{m \times n} \]

Scalar Multiples: For a scalar \(r\):

Theorem 1

Let \(A\), \(B\), and \(C\) be matrices of the same size, and let \(r\) and \(s\) be scalars.

  1. \(A + B = B + A\) (Commutative Law)
  2. \((A + B) + C = A + (B + C)\) (Associative Law)
  3. \(A + O = A\)
  4. \(r(A + B) = rA + rB\)
  5. \((r + s)A = rA + sA\)
  6. \(r(sA) = (rs)A\)

(II) Matrix Multiplication

Multiplication of matrices corresponds to the composition of linear transformations.

If \(\vec{x} \in \mathbb{R}^p\), multiplying by an \(n \times p\) matrix \(B\) yields \(B\vec{x} \in \mathbb{R}^n\). Multiplying that result by an \(m \times n\) matrix \(A\) yields \(A(B\vec{x}) \in \mathbb{R}^m\).

\[ A(B\vec{x}) = (AB)\vec{x} \]

Definition

If \(A\) is an \(m \times n\) matrix, and if \(B\) is an \(n \times p\) matrix with columns \(\vec{b}_1, \vec{b}_2, \dots, \vec{b}_p\), then the product \(AB\) is the \(m \times p\) matrix whose columns are \(A\vec{b}_1, A\vec{b}_2, \dots, A\vec{b}_p\). That is:

\[ AB = A \begin{bmatrix} \vec{b}_1 & \dots & \vec{b}_p \end{bmatrix} = \begin{bmatrix} A\vec{b}_1 & \dots & A\vec{b}_p \end{bmatrix} \]

Note: The number of columns of \(A\) must match the number of rows in \(B\) in order for a linear combination (such as \(A\vec{b}_1\)) to be defined.

Each column of \(AB\) is a linear combination of the columns of \(A\) using weights from the corresponding column of \(B\).

Row-Column Rule for Computing \(AB\)

If the product \(AB\) is defined (where \(A\) is \(m \times n\) and \(B\) is \(n \times p\)), then the \((i,j)\)-entry of \(AB\) is the sum of the products of corresponding entries from Row \(i\) of \(A\) and Column \(j\) of \(B\).

\[ (AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj} \]

This is a more efficient method for calculation.

Example 1

Use both the definition and the Row-Column Rule to compute \(AB\) for:

\[ A = \begin{bmatrix} 4 & -2 \\ -3 & 0 \\ 3 & 5 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 & 3 \\ 4 & -1 \end{bmatrix} \]

Solution using Definition:

Let \(\vec{b}_1 = \begin{bmatrix} 1 \\ 4 \end{bmatrix}\) and \(\vec{b}_2 = \begin{bmatrix} 3 \\ -1 \end{bmatrix}\).

\[ A\vec{b}_1 = \begin{bmatrix} 4(1) + (-2)(4) \\ -3(1) + 0(4) \\ 3(1) + 5(4) \end{bmatrix} = \begin{bmatrix} -4 \\ -3 \\ 23 \end{bmatrix} \] \[ A\vec{b}_2 = \begin{bmatrix} 4(3) + (-2)(-1) \\ -3(3) + 0(-1) \\ 3(3) + 5(-1) \end{bmatrix} = \begin{bmatrix} 14 \\ -9 \\ 4 \end{bmatrix} \] \[ AB = \begin{bmatrix} A\vec{b}_1 & A\vec{b}_2 \end{bmatrix} = \begin{bmatrix} -4 & 14 \\ -3 & -9 \\ 23 & 4 \end{bmatrix} \]

Solution using Row-Column Rule:

\[ AB = \begin{bmatrix} (4)(1) + (-2)(4) & (4)(3) + (-2)(-1) \\ (-3)(1) + (0)(4) & (-3)(3) + (0)(-1) \\ (3)(1) + (5)(4) & (3)(3) + (5)(-1) \end{bmatrix} = \begin{bmatrix} -4 & 14 \\ -3 & -9 \\ 23 & 4 \end{bmatrix} \]

(III) Properties of Matrix Multiplication

Theorem 2

Let \(A\) be an \(m \times n\) matrix, and let \(B\) and \(C\) have sizes for which the indicated sums and products are defined.

  1. \(A(BC) = (AB)C\) (Associative Law of Multiplication)
  2. \(A(B + C) = AB + AC\) (Left Distributive Law)
  3. \((B + C)A = BA + CA\) (Right Distributive Law)
  4. \(r(AB) = (rA)B = A(rB)\) for any scalar \(r\)
  5. \(I_m A = A = A I_n\) (Identity for Matrix Multiplication)

Warnings

  1. In general, \(AB \neq BA\) (Matrix multiplication is not commutative).
    Example: If \( A = \begin{bmatrix} -3 & 2 \\ 3 & -2 \end{bmatrix} \) and \( B = \begin{bmatrix} 2 & 0 \\ 4 & 3 \end{bmatrix} \), then: \[ AB = \begin{bmatrix} 2 & 6 \\ -2 & -6 \end{bmatrix} \neq BA = \begin{bmatrix} -6 & 4 \\ -3 & 2 \end{bmatrix} \]
  2. The Cancellation Laws do not hold for matrix multiplication.
    That is, if \(AB = AC\), then it is not true in general that \(B = C\).
    Example: Let \( A = \begin{bmatrix} -3 & 2 \\ 3 & -2 \end{bmatrix} \), \( B = \begin{bmatrix} 2 & 0 \\ 4 & 3 \end{bmatrix} \), and \( C = \begin{bmatrix} 0 & 0 \\ 1 & 3 \end{bmatrix} \).
    Here, \(AB = AC = \begin{bmatrix} 2 & 6 \\ -2 & -6 \end{bmatrix}\), but clearly \(B \neq C\).
  3. If \(AB = O\), we cannot conclude that \(A = O\) or \(B = O\) in general.
    Example: Let \( A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 4 & 2 \\ -2 & -1 \end{bmatrix} \).
    Here, \(AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O\), yet neither \(A\) nor \(B\) is the zero matrix.

Powers of a Matrix

If \(A\) is an \(n \times n\) matrix and \(k\) is a positive integer, then:

\[ A^k = A \cdot A \dots A \quad \text{(the product of } k \text{ copies of } A) \]

(IV) The Transpose of a Matrix

If \(A\) is an \(m \times n\) matrix, its transpose, denoted \(A^T\), is an \(n \times m\) matrix.

Transpose of \(A\): The columns of \(A^T\) are formed from the corresponding rows of \(A\).

Example \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \implies A^T = \begin{bmatrix} a & c \\ b & d \end{bmatrix} \]

Theorem 3

Let \(A\) and \(B\) denote matrices whose sizes are appropriate for the indicated sum and product.

  1. \((A^T)^T = A\)
  2. \((A + B)^T = A^T + B^T\)
  3. \((rA)^T = rA^T\) for any scalar \(r\)
  4. \((AB)^T = B^T A^T\)

Note on (4): The transpose of a product of matrices equals the product of their transposes in the reverse order.

Generalizing (4) for multiple matrices:

\[ (A_1 A_2 \dots A_p)^T = A_p^T \dots A_2^T A_1^T \]