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.
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Det[m] gives the determinant of a square matrix
m.
Minors[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 is
(-1)i+j times the
(n-i+1, n-j+1)th element of the matrix of minors.
Minors[m, k] 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.
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This generates a 3×3 matrix, whose i, jth entry is a[i, j].
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Here is the determinant of m.
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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.
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The
rank of a matrix is the number of linearly independent rows or columns.
This finds the rank of a matrix.
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Powers and exponentials of matrices.
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This gives the third matrix power of m.
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It is equivalent to multiplying three copies of the matrix.
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Here is the millionth matrix power.
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The matrix exponential of a matrix
m is
mk/k!, where
mk indicates a matrix power.
This gives the matrix exponential of m.
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Here is an approximation to the exponential of m, based on a power series approximation.
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