Special Matrices in Linear Algebra

hello welcome to this lesson in the linear algebra tutor up until this point we've reviewed a lot of material that should probably have been exposed to in other classes at one form or another but we're kind of recasting it in this realm of linear algebra for instance we talked about vectors a minute ago the last section and we talked about how they're really just the same concept as these guys called in tuples which are just listings of numbers and the fact that we know how to take the dot product of two vectors really just translates to the regular old matrix multiplication rule that we already know from matrix multiplication so start to get used to the idea of a vector basically being a matrix a type of matrix because in a moment we're going to be dealing a lot more with vectors in this class and that's the connection that basically vectors are simply matrices for three-dimensional space you'd have three components in that and that column matrix there now as you've probably guessed up till now a lot of linear algebra is basically dealing with definitions and making those connections clearly like we've been doing in the last section so in this lesson what we're going to do is just go over a few more special types of matrices very very simple but I want to make them bulletproof and very easy for you to understand the first one is called an identity matrix we're going to be dealing with identity matrices all throughout this class and your problems and also in proofs and things like that so let's go and talk about the concept of an identity matrix very very simple the identity matrix identity matrix the way you usually see it represented in a linear algebra book is the capital I with a subscript in you'll understand what the N means in just a second if I have an identity matrix I sub to C nose in this case n is equal to 2 then it looks like this basically it means a 2 by 2 matrix with the number 1 along the diagonal elements and zeros everywhere else all right so the identity matrix is a general term you can have an identity matrix for two by two for three by three for four by four for five by five so that's why we have the number the letter in there telling you that you can generate an identity matrix of any size but it's always going to be a square matrix so here's the identity matrix I sub to in the way that you constructive is you have ones on the diagonal zeros everywhere else now we can also have for instance an identity matrix I sub three what do you think that would look like well it's gonna be 1 0 0 0 1 0 0 0 1 notice in this case we have ones along the diagonal zeros everywhere else on the off diagonal elements and you can kind of continue on you can make I sub 4 and I sub 5 in each of those cases you'll have ones along the diagonal elements and zeros everywhere else that's when an identity matrix is so if you ever see I sub n running around a proof or a theorem and your linear algebra book just replace it with this concept of an identity matrix with ones on the diagonal now why do we care about that because an identity matrix is special because of the following reason it's pretty simple you know once we go over it'll be pretty easy for you if you take a matrix a any matrix and you multiply it by the identity matrix of the proper size the same size as a then what you're going to get is you're going to get the same matrix back now obviously the sizes of the matrices that you choose to do them here for a and I they have to be able to to generate a multiplication you know matrix multiplication we said is not always defined so you need to make sure your identity matrix is of the proper size so that you can actually do the multiplication but to give you an example of how you would do this I could generate a matrix as follows 1 3 4 2 0 1 1 1 1 right and that could be matrix a right and I can multiply it by matrix I now I can't just choose any old mate any old matrix I can't choose a two-by-two to multiply here because if I put a 2 by 2 matrix then the multiplication isn't going to be of the right dimensionality so I need to choose a 3 by 3 matrix 0 1 0 0 0 1 this is going to work because and I'm just going to put I sub 3 here this is going to work because we have we can go over and down so number of columns this direction equals the number of rows this direction so this multiplication is valid now if we actually do this let's go and do this multiplication real quick and kind of show you if we go over and down notice we have 1 times 1 give you 1 3 times 0 4 times 0 so the zeros kind of kill everything else 1 times 1 remains all right and let's just go over to the right so we'll go over and down this way so we have 1 times 0 giving you nothing 3 times 1 giving you 3 4 times 0 giving you nothing so the 3 survives and if we go over and down this way this goes gives you 0 this gives you 0 this gives you 4 so you can see what's happening is already the original matrix here is being mirrored in the answer the identity matrix is going to make it so that if you carry out this multiplication what you're going to get is the exact elements that you started with so the initial matrix a when you multiply by an identity matrix is equal to itself all right and if you want to just spot check that you can go over and down this way 1 times 1 giving you 1 0 giving you zero here this is giving you 0 here so if you go over and down this way you're gonna get the number 1 that's that guy down there so I encourage you to do the full multiplication with all the rows and all the columns and you'll see that anytime you multiply by an identity matrix for presuming that the dimensions of a and I are such that you can multiply them you're going to get the same matrix that you started with all right and that's the beauty of that how you're going to use this can't really express it too much right now because the application of when you actually need to use it will come a little bit later but I want you to understand these definitions make them clear for you all right the next thing we're gonna talk about is called triangular matrices now I want to make sure you understand when you read in a textbook or see in a lecture somebody talks about a triangle matrix I want you to understand what that is so let me give you a couple quick examples 1 0 0 2 1 0 3 4 2 all right this is called lower triangular matrix and the reason is because the nonzero elements form a lower triangle here so in other words if you had the matrix here and you have the matrix here it forms a triangle like this when the diagonal elements here are part of this line here and the other nonzero elements are inscribed in sort of a triangle now this is a three by three matrix but it also goes for larger matrices from five by five and so on a lower triangular matrix would have to basically be diagonal elements and then on down to the bottom where everything above the diagonal like this up into the right would be all zeros that's gonna be called a lower triangular matrix so you might also guess then then I can draw an upper triangular matrix one two three zero four two zero zero nine this is an upper triangular I'll use the word triangular represented by a triangle matrix and in general if you have a matrix like this and upper triangular matrix will look like something like this where this line here is the diagonal elements are nonzero we form an upper triangle here these zeros here everything on the other side of the diagonal is zero all right and then finally you can also have something just called a diagonal matrix very very simple concept to understand I'm again representing these with three by three matrices but it extends to hire in dimensionality as well zero two zero zero zero negative three this is called a diagonal matrix and a diagonal matrix basically the only nonzero elements are gonna be strictly along the diagonal like what we have right here so here we have nonzero elements on the diagonal but everything into the upper right and everything to the lower left are all zeros how are these useful usually you'll see things like this and proofs you know there are ways to take a matrix and split it into a triangular upper matrix and in lower triangular matrix and then manipulate them separately so I just want you to understand the terminology as we get into matrix algebra when you see something like triangle matrix you're not freaked out it just means basically we talked about right here now the next thing I want to mention to you is something that you'll definitely see before we talked about remember we talked about the transpose of a matrix transpose is when you take the rows of the original matrix and turn them into columns or equivalently take columns and turn them into rows basically you're flipping the rows and the columns around that's what we call the transpose we already covered that so related to that we have something called a symmetric matrix so if somebody on the test says hey tell me if this matrix is symmetric or not you need to know what to look for and the test to see if a matrix is symmetric is if the matrix if the matrix is equal to its transpose if it's equal to its transpose it's said to be symmetric all right because remember when you have a matrix you're flipping the rows and flipping the column so it's very likely in general that since you're flipping the matrix around like that what you get for the transpose will be totally different than what you start with but in a very special case if you construct a matrix properly and you take the transpose you'll actually get the same matrix that you started with that's called symmetric matrix so let's go ahead and talk about that and give you a quick example of that let's say that we have the matrix a 1 7 3 7 4 negative 5 3 negative 5 and 6 all right so how do you figure out if it's symmetric well the first thing you need to do is construct the matrix transpose and let's see if these two are equal so the way I do it is I turn the rows into columns so I go across 173 and turn that into a column and then I go here 7 4 negative 5 and turn that into a column and then I go here 3 negative 5 and 6 and I turn that into a column this is the transpose of the original matrix but notice 1 7 3 1 7 3 7 4 negative 5 7 4 95 3 negative 5 6 3 negative 5 6 the transpose of the matrix is the same as the original matrix now that's why we call it symmetric all right now in general that's not going to be the case in general that's not going to be the case because if you have a weirdly shaped matrix taking you know columns over here flipping them in rose then in general you're gonna get something different but this particular matrix is special and you can kind of see that because if you notice if you were to put like a sheet of paper right here along the diagonal you notice you have a mirror image the seven is mirrored on both sides of the diagonal the three is mirrored on both sides of the diagonal the negative five is mirrored on both sides of the diagonal and the diagonal elements themselves one four and six are actually just remaining in the same place so if you kind of take this matrix and fold it in half like a sandwich or something you're gonna end up with a mirror image across the diagonal elements that's why we call it symmetric because symmetry means you have some-some duplicity somewhere some kind of you know reflection so to speak so that's what we call symmetric matrix matrix but the test of it is you form the transpose see if you get the same thing and then you're done now the last thing that was called a symmetric matrix there's actually something called skew symmetric matrix right skew symmetric and that's very very similarly related and that is if you have a matrix a and you take its transpose then what you get is that the original matrix a is equal to negative of its transpose so it gives very similar to before it's just that the elements are they are a mirror image but they're negated relative to the original matrix so let's give a quick example and show you how to figure out if a matrix of skew symmetric or not so what we have is matrix a 0 2 negative 1 negative 2 0 negative 4 1 4 0 all right now the next step once we get our matrix is let's go ahead and form its transpose do you want things one step at a time let's go ahead and form its transpose first so I'll take this row and make it into a column 0 2 negative 1 I'll take this row and make it a column so it'd be negative 2 0 negative 4 and I'll take this and make it a column 1 4 0 all right so this is the matrix transpose and then there's a final step I'm going to take a neg the neg of transpose so the negative of that matrix that we just created notice we're multiplying the matrix by negative one so all we're doing is gonna change all the signs of the elements so be zero to one negative two zero of negative four one four and zero and that is the negative of the transpose now notice what we have zero to negative one zero two and I forgot a negative one here stick them right there notice that we have a one right here and here's a negative one so will I just forgot one there so we have a zero to negative one negative two zero negative four one four zero these all match what the original matrix was so the original matrix is equal to the negative of its transpose basically is what we what we have going on there all right and that's how you arrive if something is skew symmetric so it's pretty much symmetric I mean the absolute value of the of the transpose looking at the original matrix the same numbers are everywhere it's just that you have to put a negative sign in front to make them actually equal that's called skew symmetric so occasionally on an exam or quiz or in a homework problem you'll give you given a matrix and have to determine if it's symmetric or skew symmetric or find its transpose things like that so I just want to give you a couple quick examples to solidify that process that's what we've been doing basically here so we've just taken a few minutes to talk about some special matrices identity matrix and so I'm mainly trying to tidy up any definitions the the things that you'll read in the book and you'll need to know what what the heck they're talking about in order to follow the lesson what we're going to do in the next few sections is begin to turn our attention a little more heavily to vectors and we've done that a little bit before we've kind of made the connection between vectors and matrices and I hope that by now you have a good warm fuzzy that a vector is really just a triplet of numbers if it's a three-dimensional vector if it's two-dimensional vector it's just two numbers but in any case it can be represented as in matrix because a matrix is just a listing of numbers that's all it is and so we're going to learn how to manipulate those in terms of vectors using the rules of linear algebra so we're going to begin begin start to start to do with vectors here in in the next lesson or two and keep in mind that as we go way through the course we're going to go through a lot of topics between here and there but eventually we'll be talking a whole lot more in terms of vectors and manipulating vectors in different vector spaces so definitely don't fall asleep when we're learning about the vector stuff because the second half of linear algebra is all about vector spaces how to rotate vectors how to transform vectors how to form new vector coordinate axes and and all of that stuff so we're building up our skills and learning to treat vectors as matrices so that when we manipulate them we'll be manipulating them in the concepts of the context of linear algebra and all the transformations that are to come so let's go on to the next section we'll continue working with vectors and linear algebra

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Keywords: linear algebra, matrix, matrices, symmetric matrix, skew symmetric matrix, triangular matrix, triangular matrices, diagonal matrix, identity matrix, identity matrices, row reduction, eigenvalue, eigenvector, rank, linear, algebra, tutor, tutorial, class, linear algebra matrix, identity matrix linear algebra, matrices linear algebra, linear algebra tutorial, linear algebra practice, linear algebra lecture

Id: sI9cOlZBEOU

Length: 16min 8sec (968 seconds)

Published: Tue Nov 12 2013