Linear algebra MOC

Matrix algebra

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

1. Matrix addition — Two matrices of the same number of rows and columns are added piecewise, so for , the resulting matrix is obtained by
2. Matrix multiplication — If is and is , then the matrix product exists and is obtained from

Which more intuitively involves taking the dot product of each row of with each column of , and arranging them in a matrix such that the row position matches and the column position matches . 3. Matrix transposition — Every matrix of size has a transpose of size , such that if

The matrix transpose is closely related to Duality. For example, covectors are the transpose of vectors. 4. Scalar multiplication — Every entry of matrix is multiplied by the scalar , i.e. if

Note a matrix of size has rows and columns. Matrix multiplication algebra as a category.

Properties

From the definitions of the operations above, it follows that¹

1. (matrix addition is associative)
2. (matrix addition is associative)
3. (scalar multiplication is distributive over matrix addition)
4. (scalar multiplication is distributive over scalar addition)
5. (scalar multiplication is associative)
6. (matrix multiplication is associative)
7. and (scalar multiplication is commutative)
8. (matrix multiplication is left-distributive over addition)
9. (matrix multiplication is right-distributive over addition)
10. (transposition is an involution)
11. (transposition is distributive over addition)
12. (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.

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