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 
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[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
entry is
.
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Here is the determinant of
.
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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
.
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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 
, where 
indicates a matrix power.
This gives the matrix exponential of
.
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Here is an approximation to the exponential of
, based on a power series approximation.
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(Hermitian conjugate)
minors
matrix power
