08. Other operations
08. Other operations
Matrix addition
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 m -by-n matrices A and B with entries Aij and Bij , we can form a new matrix A+B with (A+B)ij=Aij+Bij.
(1011)+(110-1)=(2110).
Special case: vector addition
This is most useful when A and B are both column vectors, i.e. m -by-1 matrices. Let's see what it means in for vectors in 𝐑2 . The formula is (xy)+(ab)=(x+ay+b).
Geometrically, we add two vectors v=(xy) and w=(ab) by translating w to the tip of v and drawing the arrow from the tail of v to the tip of w . One can see from the picture that the x - (respectively y -) coordinate of this arrow is the sum of the x - (respectively y -) coordinates of v and w .
Rescaling
Given a number λ and a matrix A , you can form the matrix λA whose entries are λ times the entries of A .
2(1234)=(2468) .
Matrix exponentiation
The exponential of a number x is defined by the Taylor series of exp : exp(x)=1+x+x22!+x33!+⋯=∞∑n=0xnn!
Consider M=(0100) . Since M2=0 , all the higher powers of M vanish (the name for this is nilpotence: some power of M is zero), so the matrix exponential becomes exp(M)=I+M=(1101).
In fact, exp(0t00)=(1t01),
Take M=(0-tt0) . We have M=t(0-110)
and in the end we get exp(M)=I+t(0-110)