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
element gives the determinant of the submatrix obtained by deleting the
row and the
column of m
cofactor of m
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
Here is the determinant of
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
It is equivalent to multiplying three copies of the matrix.
Here is the millionth matrix power.
The matrix exponential of a matrix m
indicates a matrix power.
This gives the matrix exponential of
Here is an approximation to the exponential of
, based on a power series approximation.