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The adjugate of a diagonal matrix is again diagonal. Where all matrices are square, A matrix is diagonal if and only if it is triangular and normal. A matrix is diagonal if and only if it is both upper-and lower-triangular. A diagonal matrix is symmetric. The identity matrix I n and zero matrix are diagonal. A 1×1 matrix is always diagonal.
An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks. The eigendecomposition or diagonalization expresses A as a product VDV −1, where D is a diagonal matrix and V is a suitable invertible matrix. [52] If A can be written in this form, it is called diagonalizable.
The binary matrix with ones on the anti-diagonal, and zeroes everywhere else. a ij = δ n+1−i,j: A permutation matrix. Hilbert matrix: a ij = (i + j − 1) −1. A Hankel matrix. Identity matrix: A square diagonal matrix, with all entries on the main diagonal equal to 1, and the rest 0. a ij = δ ij: Lehmer matrix: a ij = min(i, j) ÷ max(i, j).
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
The identity matrix commutes with all matrices. Jordan blocks commute with upper triangular matrices that have the same value along bands. If the product of two symmetric matrices is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices. [9] [10] Circulant matrices commute.
For example, we can conveniently require the lower triangular matrix L to be a unit triangular matrix, so that all the entries of its main diagonal are set to one. Then the system of equations has the following solution:
The set of Toeplitz matrices is a subspace of the vector space of matrices (under matrix addition and scalar multiplication). Two Toeplitz matrices may be added in O ( n ) {\displaystyle O(n)} time (by storing only one value of each diagonal) and multiplied in O ( n 2 ) {\displaystyle O(n^{2})} time.