Knowlpedia

Matrix

A rectangular array of numbers, symbols, or expressions arranged in rows and columns.

A matrix is a rectangular array of entries (typically from a or ) arranged in rows and columns.

An m×nm \times n matrix AA has mm rows and nn columns, written

A=(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn).A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}.

The entry in row ii and column jj is denoted aija_{ij} or AijA_{ij}.

Operations
  • Addition: (A+B)ij=Aij+Bij(A + B)_{ij} = A_{ij} + B_{ij} (requires same dimensions).
  • Scalar multiplication: (cA)ij=c⋅Aij(cA)_{ij} = c \cdot A_{ij}.
  • Matrix multiplication: (AB)ij=∑kAikBkj(AB)_{ij} = \sum_k A_{ik} B_{kj} (requires compatible dimensions).
Special matrices
  • Square matrix: m=nm = n.
  • Identity matrix: InI_n with 11s on diagonal, 00s elsewhere.
  • Zero matrix: All entries are 00.
  • Diagonal matrix: Every entry off the main diagonal is zero.
  • Symmetric matrix: A=ATA = A^T (equals its transpose).
Remarks

Matrices represent between finite-dimensional once bases are chosen.

Transposition and symmetry

The interchanges row and column indices. A equals its transpose.