This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 3.7.7 Basic Matrix Operations Some basic matrix operations. Transposing a matrix interchanges the rows and columns in the matrix. If you transpose an matrix, you get an matrix as the result. Transposing a matrix gives a result. In[1]:= Transpose[ {{a, b, c}, {ap, bp, cp}} ] Out[1]= Det[m] gives the determinant of a square matrix m. Minors[m,k] gives a matrix of the determinants of all the submatrices of m. You can apply Minors to rectangular, as well as square, matrices. Here is the determinant of a simple matrix. In[2]:= Det[ {{a, b}, {c, d}} ] Out[2]= This generates a matrix, whose entry is a[ i,j]. In[3]:= m = Array[a, {3, 3}] Out[3]= Here is the determinant of m. In[4]:= Det[ m ] Out[4]= This gives the matrix of all minors of m. In[5]:= Minors[m, 2] Out[5]= You can use Det to find the characteristic polynomial for a matrix. Section 3.7.9 discusses ways to find eigenvalues and eigenvectors directly. Here is a matrix. In[6]:= m = Table[ 1/(i + j), {i, 3}, {j, 3} ] Out[6]= Following precisely the standard mathematical definition, this gives the characteristic polynomial for m. In[7]:= Det[ m - x IdentityMatrix[3] ] Out[7]= There are many other operations on matrices that can be built up from standard Mathematica functions. One example is the trace or spur of a matrix, given by the sum of the terms on the leading diagonal. Here is a simple matrix. In[8]:= m = {{a, b}, {c, d}} Out[8]= You can get the trace of the matrix by explicitly constructing a sum of the elements on its leading diagonal. In[9]:= Sum[ m[[i, i]], {i, 2} ] Out[9]= Powers and exponentials of matrices. Here is a matrix. In[10]:= m = {{0.4, 0.6}, {0.525, 0.475}} Out[10]= This gives the third matrix power of m. In[11]:= MatrixPower[m, 3] Out[11]= It is equivalent to multiplying three copies of the matrix. In[12]:= m . m . m Out[12]= Here is the millionth matrix power. In[13]:= MatrixPower[m, 10^6] Out[13]= This gives the matrix exponential of m. In[14]:= MatrixExp[m] Out[14]= Here is an approximation to the exponential of m, based on a power series approximation. In[15]:= Sum[MatrixPower[m, i]/i!, {i, 0, 5}] Out[15]=