08. Other operations
In this video, we'll define some further operations you can do to produce new matrices. The first is matrix addition If we have two -by- matrices and with entries and , we can form a new matrix with In other words, you take the th entries of both matrices and add them.
Special case: vector addition
This is most useful when and are both column vectors, i.e. -by- matrices. Let's see what it means in for vectors in . The formula is
Geometrically, we add two vectors and by translating to the tip of and drawing the arrow from the tail of to the tip of . One can see from the picture that the - (respectively -) coordinate of this arrow is the sum of the - (respectively -) coordinates of and .
Given a number and a matrix , you can form the matrix whose entries are times the entries of .
The exponential of a number is defined by the Taylor series of : We can use the same definition to define the exponential of a matrix: Here, is understood to mean the identity matrix (the analogue for matrices of the number ).
Consider . Since , all the higher powers of vanish (the name for this is nilpotence: some power of is zero), so the matrix exponential becomes So we get the matrix for a shear as the exponential of a nilpotent matrix.
In fact, so we get a whole family of matrices which shear further and further to the right as varies.
Take . We have etc.
and in the end we get etc. The coefficient of is the Taylor series for ; the coefficent of is the Taylor series for , so overall we get So we get a general rotation matrix in 2-d by exponentiating this very simple matrix.