Matrix algebra

A matrix algebra involves rectangular arrays with entries from some Field $𝕂$ , and the following operations

Matrix addition — Two matrices of the same number of rows and columns are added piecewise, so for $𝐶 = 𝐴 + 𝐵$ , the resulting matrix is obtained by $𝑐 𝑖 𝑗 = 𝑎 𝑖 𝑗 + 𝑏 𝑖 𝑗$
Matrix multiplication — If $𝐴 = (𝑎 𝑖 𝑗)$ is $𝑚 \times 𝑝$ and $𝐵 = (𝑎 𝑖 𝑗)$ is $𝑝 \times 𝑛$ , then the matrix product $𝐴 𝐵$ exists and is obtained from

𝑐 𝑖 𝑗 = 𝑘 = 𝑝 \sum 𝑘 = 1 𝑎 𝑖 𝑘 𝑏 𝑘 𝑘

Which more intuitively involves taking the dot product of each row of $𝐴$ with each column of $𝐵$ , and arranging them in a matrix such that the row position matches $𝐴$ and the column position matches $𝐵$ . 3. Matrix transposition — Every matrix $𝐴$ of size $𝑚 \times 𝑛$ has a transpose $𝐴 𝖳$ of size $𝑛 \times 𝑚$ , such that if $𝐴 𝖳 = (𝑐 𝑖 𝑗)$

𝑐 𝑗 𝑖 = 𝑎 𝑖 𝑗

The matrix transpose is closely related to Duality. For example, covectors are the transpose of vectors. 4. Scalar multiplication — Every entry of matrix $𝐴 = (𝑎 𝑖 𝑗)$ is multiplied by the scalar $𝛼$ , i.e. if $𝛼 𝐴 = (𝑐 𝑖 𝑗)$

𝑐 𝑖 𝑗 = 𝛼 𝑎 𝑖 𝑗

Note a matrix of size $𝑚 \times 𝑛$ has $𝑚$ rows and $𝑛$ columns. Matrix multiplication algebra as a category.

Properties

From the definitions of the operations above, it follows that¹

$𝐴 + 𝐵 = 𝐵 + 𝐴$ (matrix addition is associative)
$(𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶)$ (matrix addition is associative)
$𝛼 (𝐴 + 𝐵) = 𝛼 𝐴 + 𝛼 𝐵$ (scalar multiplication is distributive over matrix addition)
$(𝛼 + 𝛽) 𝐴 = 𝛼 𝐴 + 𝛽 𝐴$ (scalar multiplication is distributive over scalar addition)
$(𝛼 𝛽) 𝐴 = 𝛼 (𝛽 𝐴)$ (scalar multiplication is associative)
$𝐴 (𝐵 𝐶) = (𝐴 𝐵) 𝐶$ (matrix multiplication is associative)
$(𝛼 𝐴) 𝐵 = 𝛼 (𝐴 𝐵)$ and $𝐴 (𝛼 𝐵) = 𝛼 (𝐴 𝐵)$ (scalar multiplication is commutative)
$𝐴 (𝐵 + 𝐶) = 𝐴 𝐵 + 𝐴 𝐶$ (matrix multiplication is left-distributive over addition)
$(𝐴 + 𝐵) 𝐶 = 𝐴 𝐶 + 𝐵 𝐶$ (matrix multiplication is right-distributive over addition)
$(𝐴 𝖳) 𝖳$ (transposition is an involution)
$(𝐴 + 𝐵) 𝖳 = 𝐴 𝖳 + 𝐵 𝖳$ (transposition is distributive over addition)
$(𝐴 𝐵) 𝖳 = 𝐵 𝖳 𝐴 𝖳$ (transposition is anti-distributive over square matrix multiplication)

Notable differences between matrix algebra and the real numbers are

Matrix multiplication is not distributive
The multiplicative identity, and therefore the multiplicative inverse, only exist in square matrices. See Types of square matrix.

#state/tidy | #SemBr

2022. MATH1012: Mathematical theory and methods, pp. 52n. ↩

Matrix algebra

Properties

Footnotes