Matrices

A matrix is a representation of a linear system, so the system

Is the matrix

In addition, a matrix is a "rule" for a linear transformation of a vector, and the matrix-vector product is applying a transformation to said vector.

A matrix is a rectangular array with rows and columns. The entry in the th row and th column is denoted by .

Which is abbreviated as when the size is known. The set of all matrices with real entries is denoted by .

Two matrices are equal if for all

Let . If for all , then

See vectors

A matrix is square is . That is, it has the same number of rows and columns.

The matrix with zero entries is called the zero matrix, denoted by , or sometimes just if the size is clear.

A matrix is symmetric if . That is, if its transpose is the same as itself.

To be symmetrical, a matrix must be a square matrix.

Denoted by or or , is the square matrix with size with for .

for any

Example

(See Matrix-Vector Product)

Let . is called upper triangular if every entry below the main diagonal is zero. is called lower triangular if every entry above the main diagonal is zero.

Upper triangular matrices:

Lower triangular matrices: