Matrix
A matrix is a rectangular array of numbers supporting addition, subtraction, multiplication, transposition, and inversion; it is the foundation of linear algebra and computer graphics.
Purpose
To represent and manipulate linear transformations and linear systems, central to scientific and engineering computing.
Mechanism
An m×n matrix has m rows and n columns. Multiplication: (AB)[i][j]=ΣA[i][k]B[k][j]. Transposition swaps rows and columns. The inverse A^(-1) satisfies AA^(-1)=I. The determinant det measures invertibility. The Strassen algorithm accelerates multiplication.
Use cases
- 3D graphics transformations
- Linear system solving
- Machine learning (weight matrices)
- PageRank computation
Examples
- Input
[[1,2],[3,4]] × [[5,6],[7,8]]Output[[19,22],[43,50]]Matrix multiplication
Common errors
- Assuming matrix multiplication commutes
- AB ≠ BA; mind the order
Related terms
Vector
A vector is a quantity with magnitude and direction, represented as an ordered array of numbers, supporting addition, scalar multiplication, dot products, and cross products; it is the foundation of physics and geometry. GLOSSARY
Complex Number
A complex number has the form a+bi (i²=-1), extending the reals to two dimensions; it underpins signal processing and quantum computing. GLOSSARY
Polynomial
A polynomial is a linear combination of powers of a variable (e.g., ax²+bx+c), supporting addition, subtraction, multiplication, division, differentiation, and integration; it is the foundation of algebra and numerical computing. GLOSSARY