4.1. Matrices#

Definition 4.1 (Matrix)

An \(m \times n\) matrix \(\bA\) is a rectangular array of numbers.

\[\begin{split} \bA = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn}\\ \end{bmatrix}. \end{split}\]

The numbers in a matrix are called its elements.

The matrix consists of \(m\) rows and \(n\) columns. The entry in \(i\)-th row and \(j\)-th column is referred with the notation \(a_{ij}\).

If all the elements of a matrix are real, then we call it a real matrix.

If any of the elements of the matrix is complex, then we call it a complex matrix.

A matrix is often written in short as \(\bA = (a_{ij})\).

Definition 4.2 (The set of matrices)

The set of all real matrices of shape \(m \times n\) is denoted by \(\RR^{m \times n}\).

The set of all complex matrices of shape \(m \times n\) is denoted by \(\CC^{m \times n}\).

Definition 4.3 (Vector)

A vector is an \(n\)-tuple of numbers written as:

\[ \bv = (v_1, v_2, \dots, v_n). \]

If all the numbers are real, then it is called a real vector belonging to the set \(\RR^n\).

If any of the numbers is complex, then it is called a complex vector belonging to the set \(\CC^n\).

The numbers in a vector are called its components.

Definition 4.4 (Column vector)

A matrix with shape \(m \times 1\) is called a column vector.

Definition 4.5 (Row vector)

A matrix with shape \(1 \times n\) is called a row vector.

Note

It should be easy to see that \(\RR^{m \times 1}\) and \(\RR^m\) are same sets. Similarly, \(\RR^{1\times n}\) and \(\RR^n\) are same sets.

A row or column vector can easily be written as an \(n\)-tuple.

Definition 4.6 (Matrix addition)

Let \(\bA\) and \(\bB\) be two matrices with same shape \(m \times n\). Then, their addition is defined as:

\[ \bA + \bB = (a_{ij}) + (b_{ij}) \triangleq (a_{ij} + b_{ij}). \]

Definition 4.7 (Scalar multiplication)

Let \(\bA\) be a matrix of shape \(m \times n\) and \(\lambda\) be a scalar. The product of the matrix \(\bA\) with the scalar \(\lambda\) is defined as:

\[ \lambda \bA = \bA \lambda \triangleq (\lambda a_{ij}). \]

Theorem 4.1 (Properties of matrix addition and scalar multiplication)

Let \(\bA, \bB, \bC\) be matrices of shape \(m \times n\). Let \(\lambda, \mu\) be scalars. Then:

  1. Matrix addition is commutative: \(\bA + \bB = \bB + \bA\).

  2. Matrix addition is associative: \(\bA + (\bB + \bC) = (\bA + \bB) + \bC\).

  3. Addition in scalars distributes over scalar multiplication: \((\lambda + \mu)\bA = \lambda \bA + \mu \bA\).

  4. Scalar multiplication distributes over addition of matrices: \(\lambda (\bA + \bB) = \lambda \bA + \lambda \bB\).

  5. Multiplication in scalars commutes with scalar multiplication: \((\lambda \mu) \bA = \lambda (\mu \bA)\).

  6. There exists a matrix with all elements being zero denoted by \(\ZERO\) such that \(\bA + \ZERO = \ZERO + \bA = \bA\).

  7. Existence of additive inverse: \(\bA + (-1)\bA = \ZERO\).

Definition 4.8 (Matrix multiplication)

If \(\bA\) is an \(m \times n\) matrix and \(\bB\) is an \(n \times p\) matrix (thus, \(\bA\) has same number of columns as \(\bB\) has rows), then we define the product of \(\bA\) and \(\bB\) as:

\[ \bA \bB \triangleq \left ( \sum_{k=1}^n a_{ik} b_{kj} \right ). \]

This binary operation is known as matrix multiplication. The product matrix has the shape \(m \times p\). Its \(i,j\)-th element is \(\sum_{k=1}^n a_{ik} b_{kj}\) obtained by multiplying the \(i\)-th row of \(A\) with the \(j\)-th column of \(B\) element by element and then summing over them.

Theorem 4.2 (Properties of matrix multiplication)

Let \(\bA, \bB, \bC\) be matrices of appropriate shape.

  1. Matrix multiplication is associative: \(\bA (\bB \bC) = (\bA \bB)\bC\).

  2. Matrix multiplication distributes over matrix addition: \(\bA (\bB + \bC) = \bA \bB + \bA \bC\) and \((\bA + \bB) \bC = \bA \bC + \bB \bC\).