Inner & outer products | Lecture 5 | Matrix Algebra for Engineers

so in this video I want to talk about inner and outer products it's a good time to introduce these because now we know about the transpose operator so let's start with the inner product this is the inner product between two vectors so we have two vectors let's say here we're going to use column vectors which is standard for matrix algebra so let's call the first vector U and that will have we'll do three rows one column three by one so this is mu 1 mu 2 mmm and the second vector will call V which is V 1 V 2 V 3 so the inner product is the same thing as the dot product if you've learned the dot product before so the dot product between these two vectors or the inner product should be U 1 V 1 plus u 2 V 2 plus u 3 V 3 how do we do that in matrix algebra well we can use the transpose operator so if we write u transpose times V then the transpose of a column vector is a row vector so we have u 1 u 2 u 3 times V which is V 1 V 2 3 so here we have a 1 row 3 columns so 1 by 3 times 3 by 1 and we end up with a 1 by 1 which is what we call a scalar so this is a straightforward multiplication u 1 V 1 plus u 2 mu 2 plus 3 okay we have some definitions here that are use useful so if this turned out to be 0 so if you and u transpose V equals law then we say that U and V are orthogonal I hope so that word we use in matrix algebra is orthogonal this is the same meaning as perpendicular for the dot product okay we have another definition so we we talk about the norm of a vector this is represented say the norm of the what do you is written with these double absolute value signs you think of this as the length of U this is given in terms of matrices is u transpose times u that's a scalar right and we Rosal to the one-half power which means taking the square root so this is just you you this is just you won't squared plus u 2 squared plus u 3 squared all to the one-half power or to the square root okay one more one more word we use we say that U is normalized so we say u is normal lowest if the norm of U is equal to 1 okay so we say vectors are normalized if their norms are one so usually you do that you normalize a vector by dividing by this by the norm of the vector and then the vector is normalized okay then there's another terminology if two vectors are orthogonal plus normalized okay then so that means we have two vectors U and V they're orthogonal so u transpose V equals zero and they're normalized so the norm of U is 1 and the norm of V is 1 then you say that the vectors are also normal okay so that's a word that is used very frequently in matrix algebra ok so this is all about inner products let's see what is the outer product the inner product is used all the time the outer product is not used really used that often but there are some numerical methods there are some techniques that make use of the outer product so as a student in matrix algebra you should know what an outer product is the inner product between two vectors was u transpose V for two column vectors that gave us a scalar but what would happen if we did u V transpose so what would you he transpose look like okay so now u is a column vector right so u 1 u 2 u 3 and V transpose then is a row vector so that would be V 1 V 2 V 3 so it's this funny multiplication right so what is this this is 3 by 1 right 3 rows one column 3 by 1 this is 1 by 3 1 row 3 columns 3 by 1 times 1 by 3 is 3 by 3 so this is a 3 by 3 matrix okay rather odd looking so we can calculate it so go across the first row down the first column kind of boring right there's only one hell in the first row and one element in the first column so that's u 1 u 1 u 1 V 2 u 1 V thing right and then the second row first column u 2 V 1 u 2 V 2 until V for me and then finally you 3 u 1 u 3 V 2 ok 3x3 matrix it's a rather strange matrix the first row has u 1 the second row has you two the third row has u 3 the first column has V 1 the first column as V 2 the first column that is V 3 well we'll see that this this type of matrix in some sense lives in a very low dimensional space but we'll talk about that more when we talk about more advanced topics in matrix algebra okay so let's recap we're talking about inner and outer product spaces here so the inner product between two column matrices is a u transpose V that gives us a scalar that's equivalent to the dot product in vector calculus if u transpose V equals zero then we say the two vectors are orthogonal the norm of the vector which is equivalent to the length of the vector is U transpose use u raised to the 1/2 power so square root of the sum of the squares of the components we say u is normalized if the norm is equal to 1 and if we have a set of vectors that are mutually orthogonal and all normalized to norm of 1 then we say the set of vectors R is orthonormal okay so that's this important word orthonormal we also talked about an outer product which is a funny matrix as UV transpose and that will for three for a column vector that's three by one that works out two or three by three matrix okay this has some more less use but it's still interesting vector product I'm Jeff jazz Knopf thanks for watching and I'll see you in the next video you

Channel: Jeffrey Chasnov

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Keywords: linear algebra, matrix algebra, matrices, inner product, outer product, inner product of matrices, outer product of matrices, matrices inner product, matrices outer product

Id: FCmH4MqbFGs

Length: 9min 26sec (566 seconds)

Published: Mon Jul 09 2018