4x4matris

A 4x4matris, or simply a 4x4 matrix, is a square array consisting of four rows and four columns, containing 16 entries a_ij organized in a grid. It is commonly denoted by A and written as A = [a_ij] with i and j ranging from 1 to 4. The elements may be real or complex, depending on the context.

The determinant det(A) is a scalar function of the entries; a matrix is invertible if det(A) ≠ 0, in which case it has a unique inverse A^{-1} satisfying AA^{-1} = A^{-1}A = I4. The trace tr(A) is the sum of the diagonal entries a_11 + a_22 + a_33 + a_44. The eigenvalues of A are scalars λ satisfying det(A − λI) = 0.

4x4matriser can represent linear transformations from R^4 to R^4. They can be multiplied to compose transformations and applied to column vectors. Basic operations include addition, scalar multiplication, and multiplication of two 4x4 matrices. The transpose A^T is obtained by swapping rows and columns.

Special forms include the identity matrix I4, diagonal matrices diag(d1,d2,d3,d4), and triangular matrices such as upper or lower triangular. In many applications, the matrix is used in a form that shows a 3×3 submatrix and an extra row/column for homogeneous coordinates. In computer graphics, a common convention uses homogeneous coordinates so a 4x4 matrix represents a 3D affine transformation with a top-left 3×3 submatrix for rotation/scale and a rightmost column for translation, while the bottom row is [0 0 0 1].

In summary, a 4x4matris is a fundamental object in linear algebra with wide applications in mathematics, science, and computer graphics.

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