1 Diagonal matrices
Matrix is diagonal if all its off-diagonal elements are zero:
where are the diagonal elements of . For simplicity, off-diagonal elements are often omitted.
A diagonal matrix where all diagonal elements are zero: A diagonal matrix where all diagonal elements are one: A diagonal matrix with constant diagonal elements:
Diagonal matrices have several important properties:
11 Linear independence
The fundamental property of diagonal matrices is linear independence of their columns, which holds if and only if all diagonal elements are non-zero. This directly follows from the fact that any linear combination of columns equaling zero requires all coefficients to be zero.
For any two distinct columns of a diagonal matrix , their dot product is zero since they have non-overlapping non-zero elements:
12 Basis set
For a diagonal matrix with non-zero diagonal elements, the columns form an orthogonal basis. Each column contains exactly one non-zero element :
Diagonal matrices are well suited for storing bases. If all diagonal elements are equal to one, the matrix is an identity matrix and it stores orthonormal basis vectors of the standard basis. In 3D space, the identity matrix stores the standard basis vectors:
This orthogonal basis has a clear geometric interpretation — each basis vector aligns with a coordinate axis and has magnitude . When all , the basis becomes orthonormal.
13 Scaling
A diagonal matrix performs scaling transformations by independently scaling each coordinate by its corresponding diagonal element:
This represents stretching or compressing the space along each coordinate axis by factors .
14 Inverse matrix
For a diagonal matrix, its inverse is obtained by taking the reciprocal of each diagonal element, provided all diagonal elements are non-zero.
Assuming, that the inverse of a diagonal matrix is , then:
15 Commutativity
Moreover, diagonal matrices form a commutative group under multiplication. Diagonal matrices have the special property that they commute under multiplication:
For two diagonal matrices and :
Any diagonal matrix can be decomposed into a product of diagonal matrices, at least:
Therefore, any order of multiplication of diagonal matrices results in the same diagonal matrix:
16 Eigenvalues and eigenvectors
For a diagonal matrix, the eigenvalues are precisely its diagonal elements, while the eigenvectors are the standard basis vectors of the space.
Eigenvalues and eigenvectors arise from the matrix equation , where is a square matrix, is an eigenvalue, and is an eigenvector. When viewing as an operator, eigenvectors represent directions that maintain their orientation under the operation, being only scaled by the factor .
For a standard basis vector :
so, is an eigenvector of with eigenvalue .