 So let a be a square matrix. Let's say it's in my end We say that a is a nil potent matrix if a race to the nth power is equal to the zero Matrix here. So some power of the matrix is equal to zero. That's actually where this word nil potent comes from It basically means zero power Some power of this matrix will give you zero. It's gonna be the nth power right here Now it turns out that with a nil potent matrix It's actually very possible that some smaller power than n will do it to you For example, we can construct a three by three matrix, which is nil potent, which would mean a cubed is equal to zero But it could also be possible that some smaller power that in fact a squared might actually be equal to zero For example the zero matrix. Let's take the three by three zero matrix It's actually nil potent because it's already equal to zero if you take the cube you're gonna get zero still So the matrix zero zero zero zero zero zero zero zero zero zero zero zero zero the zero matrix is nil potent Because when you cube it you get back the zero matrix although you get away with the first power So oftentimes when we have nil potent matrices We are curious like what's the smallest power that we can get away with that causes it to be zero Now in the previous video We actually use the outer product to construct ident potent matrices matrices such that a squared equals a Using nil potent excuse me also using the outer product We can construct nil potent matrices with the property that a squared equals zero now I don't want you to think that every nil potent matrix squares to be zero But the outer product to be used to construct nil potent matrices which square to be zero and you're actually going to use Orthogonal vectors to do that. So imagine we have two vectors u and v which are both in fn right here And suppose they're orthogonal therefore u times v u dot v is equal to zero Well, then if we construct an nil potent matrix U tensor v so we're using the outer product now instead of the inner product Then this will be a nil potent matrix with the condition that a squared equals zero to see this It's kind of a fun little argument right here notice that if I take a squared This is going to equal u tensor v times u tensor v which The the tensor product there the outer product has become u v transpose times u v transpose and when you put that together you can redo the parentheses you're going to get u v transpose Times u times v transpose. This is going to equal if you look at the middle perspective This is going to be the same thing as v dot u Times v transpose. So that's just the that's just the inner product there and because the matrix the two vectors are orthogonal The inner product is zero. So we're going to get u times zero times v transpose And so at any point, you know, you get zero in there the whole thing is going to become zero, right? I and so this is going to turn out to be the zero matrix The zero matrix when we're done. So in fact a squared was equal to zero. This is an example of a nil potent matrix Let's show you how this thing works in practice. So let's take two vectors u equals one two and v equals two negative one It's very easy to see that this in fact is these are orthogonal here You're going to get two minus two which equals zero and So if we take the outer product of these two matrices of these two vectors, excuse me So if we take one two times two negative one You're going to get one times two which is two you're going to get one times negative one Which is negative one you're going to get two times two which is four And then you're going to get two times negative one which is equal to negative two so this is the outer product of the two vectors and I claim that this is in fact a nil potent matrix if we square this thing you take the first row times the first column You're going to get two times two which is four minus four, which is zero take the second column You're going to get negative two plus and I will minus a negative two. It's a double negative there You had negative two plus two which is zero Keep on going here. You're gonna get four negative two times two four You're gonna get eight minus eight which is zero and then finally you get the Second row second column you're gonna get negative four plus four which is zero and so this is what you get here You get four different zeros in four different ways, right? But a square does turn out to be zero so we can construct nil potent matrices using the outer product This is not the only way one can construct a nil potent matrix I do want to make mention that nil potent matrices They're actually very easy to construct using strictly upper triangular matrices So for example, if you take the matrix zero one zero zero I want you to convince yourself that this is a nil potent matrix. You end up with zero zero zero zero Okay, and you could construct this you could construct this using a Outer product right here another example of you do three by three matrices. So we want this to be strictly upper triangular You could do something like zero one zero This would be an example of the matrix of squares to be zero That's perfectly fine. And again, you could construct this using it in an outer product But let me give you the example right here if I take one one zero This matrix is nil potent, but if you were to multiply together That is if you square the matrix You're not gonna get zero right here. Notice what happens when you take the first row first column You're gonna get zero great first row second column you end up with zero First row third column this one you actually do have a one that matches up You're gonna get a one right here next if you take the second row first column You're gonna get a zero second row second column You also get zero second row third column you end up a zero and then everything else is gonna be zero right here So notice what happened here is that you know analyzing this thing if you started off with like zeros One away from the diagonal entry when we squared it actually kind of pushed this these ones up by one to the corner You're kind of squishing it into the corner there and so this time if we take the matrix Zero one zero zero zero one zero zero zero zero if you times this by then it's square which is zero zero one zero zero zero zero zero zero like so in this situation You're gonna see that the product is in fact now all zeros. I'll let you double check that on your own So this matrix right here. Let's say that a is equal to this matrix zero one zero zero one zero zero zero This is an example of a nil potent matrix where a squared is not zero, but a cubed is zero So it is possible to construct nil potent matrices that The square is non-zero, but some higher power is and the easiest way to construct these nil potent matrices is strictly is Honestly speaking using these strictly upper triangular matrices every every strictly upper triangular matrix or strictly lower triangular matrix will be Nil potent and based upon how many numbers you have above the diagonal that are non-zero will determine how many powers it takes But if you want to construct some non triangular nil potent matrices, the outer product is a cool trick that you can do