Basic Matrix Operations
Some basic matrix operations.
Transposing a matrix interchanges the rows and columns in the matrix. If you transpose an
m×
n matrix, you get an
n×
m matrix as the result.
Transposing a 2×3 matrix gives a 3×2 result.
| Out[1]= |  |
Det[m] gives the determinant of a square matrix
m.
Minors[m] is the matrix whose
element gives the determinant of the submatrix obtained by deleting the
row and the
column of
m. The
cofactor of
m is
times the
element of the matrix of minors.
Minors
gives the determinants of the
k×
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.
| Out[2]= |  |
This generates a 3×3 matrix, whose
entry is
.
| Out[3]= |  |
Here is the determinant of
.
| Out[4]= |  |
The
trace or
spur 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.
| Out[5]= |  |
The
rank of a matrix is the number of linearly independent rows or columns.
This finds the rank of a matrix.
| Out[6]= |  |
Powers and exponentials of matrices.
| Out[7]= |  |
This gives the third matrix power of
.
| Out[8]= |  |
It is equivalent to multiplying three copies of the matrix.
| Out[9]= |  |
Here is the millionth matrix power.
| Out[10]= |  |
The matrix exponential of a matrix
m is
, where
indicates a matrix power.
This gives the matrix exponential of
.
| Out[11]= |  |
Here is an approximation to the exponential of
, based on a power series approximation.
| Out[12]= |  |
(Hermitian conjugate)
minors
matrix power
