Basic Operations on Matrices

Matrices support the operations of addition, scalar multiplication, transposition, and matrix multiplication. Adjacent to these are the derived operations — subtraction, raising to a power, and the trace of a matrix.

Addition of Matrices

Definition. Only matrices of the same size can be added. The sum of matrices $A = (a_{ij})$ and $B = (b_{ij})$ of size $m \times n$ is the matrix $C = A + B$ of the same size, whose entries are

c_{ij} = a_{ij} + b_{ij} .

Example:

(1324) + (05 - 1 2) = (1816) .

Multiplying a Matrix by a Number

Definition. The product of a matrix $A = (a_{ij})$ by a number (scalar) $λ$ is the matrix $λ A$ , in which every entry is multiplied by $λ$ :

λ \cdot A = λ a_{11} λ a_{21} ⋮ λ a_{m 1} λ a_{12} λ a_{22} ⋮ λ a_{m 2} \dots \dots ⋱ \dots λ a_{1 n} λ a_{2 n} ⋮ λ a_{mn} .

Example:

3 \cdot (10 - 2 4) = (30 - 6 12) .

Subtraction of Matrices

Subtraction is defined via addition with the opposite matrix:

A - B = A + (- 1) \cdot B, (A - B)_{ij} = a_{ij} - b_{ij} .

Combined with addition and scalar multiplication, this gives a linear combination of matrices $λ A + μ B$ (of the same size).

Transposing a Matrix

Definition. The transpose of a matrix $A$ is the operation of swapping rows for columns; the result is denoted $A^{T}$ . If $A = (a_{ij})$ has size $m \times n$ , then $A^{T} = (a_{j i})$ has size $n \times m$ .

Example:

A = (132405), A^{T} = 120345 .

(For the properties of transposition, see the article Properties of Matrix Operations.)

Matrix Multiplication

Definition. The product $A \cdot B$ is defined only when the number of columns of the first matrix equals the number of rows of the second (the matrices are conformable). If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, the result $C = A \cdot B$ is an $m \times p$ matrix whose entries are given by "row times column" — the dot product of the $i$ -th row of $A$ and the $j$ -th column of $B$ :

c_{ij} = k = 1 \sum n a_{ik} b_{k j} .

m \times n A \cdot n \times p B = m \times p C (the inner dimensions n must match) .

General form for $2 \times 2$ :

A = (a_{11} a_{21} a_{12} a_{22}), B = (b_{11} b_{21} b_{12} b_{22}),

C = A \cdot B = (a_{11} b_{11} + a_{12} b_{21} a_{21} b_{11} + a_{22} b_{21} a_{11} b_{12} + a_{12} b_{22} a_{21} b_{12} + a_{22} b_{22}) .

Numerical example:

(1324) (5768) = (1 \cdot 5 + 2 \cdot 7 3 \cdot 5 + 4 \cdot 7 1 \cdot 6 + 2 \cdot 8 3 \cdot 6 + 4 \cdot 8) = (19432250) .

An example with a size change ( $2 \times 3$ by $3 \times 2$ gives $2 \times 2$ ):

(1 - 1 0321) 401120 = (6 - 3 15) .

Remark. Matrix multiplication is not commutative: in general $A \cdot B \neq = B \cdot A$ (moreover, one of the products may not even be defined). For more, see the article Properties of Matrix Operations.

Raising to a Power

For a square matrix $A$ , nonnegative integer powers are defined:

A^{0} = E, A^{k} = k times A \cdot A \dots A (k \in N) .

Trace of a Matrix

Definition. The trace of a square matrix $A_{n \times n}$ is the sum of the entries on its main diagonal:

tr A = a_{11} + a_{22} + \dots + a_{nn} = i = 1 \sum n a_{ii} .

Example:

A = (2175), tr A = 2 + 5 = 7.

The trace is linear ( $tr (λ A + μ B) = λ tr A + μ tr B$ ) and has the cyclic property $tr (A B) = tr (B A)$ . The trace will come in handy when studying eigenvalues: it equals their sum.

Addition of Matrices

Multiplying a Matrix by a Number

Subtraction of Matrices

Transposing a Matrix

Matrix Multiplication

Raising to a Power

Trace of a Matrix

Related works