Basic Matrix Operations
Some basic matrix operations.
Transposing a matrix interchanges the rows and columns in the matrix. If you transpose an m
matrix, you get an n
matrix as the result.
Transposing a 2×3 matrix gives a 3×2 result.
gives the determinant of a square matrix m
is the matrix whose (i, j)th
element gives the determinant of the submatrix obtained by deleting the (n-i+1)th
row and the (n-j+1)th
column of m
. The (i, j)th
cofactor of m
times the (n-i+1, n-j+1)th
element of the matrix of minors.
gives the determinants of the k
submatrices obtained by picking each possible set of k
rows and k
columns from m
. Note that you can apply Minors
to rectangular, as well as square, matrices.
Here is the determinant of a simple 2×2 matrix.
This generates a 3×3 matrix, whose i, jth
entry is a[i, j]
Here is the determinant of m
of a matrix Tr[m]
is the sum of the terms on the leading diagonal.
This finds the trace of a simple 2×2 matrix.
of a matrix is the number of linearly independent rows or columns.
This finds the rank of a matrix.
Powers and exponentials of matrices.
This gives the third matrix power of m
It is equivalent to multiplying three copies of the matrix.
Here is the millionth matrix power.
The matrix exponential of a matrix m
, where mk
indicates a matrix power.
This gives the matrix exponential of m
Here is an approximation to the exponential of m
, based on a power series approximation.