what is a diagonal matrix
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in this video lets see another type of matrix called as the diagonal matrix we have already seen in the previous videos three important types of matrices call as the row matrix the column matrix and square matrix and in this video is the continuation to the previous ones here we are going to study what is known as the diagonal matrix now lets see what is the definition of diagonals I have already told you in the square matrix what do you mean by the principal diagonal or the leading diagonal that knowledge will be helpful in this video of diagonal matrix right diagonal matrix is what a square matrix a I J n by n that means the square matrix of order n is called a diagonal matrix if all elements except the ones in the leading diagonal are 0 I repeat it again that means diagonal matrix is basically an extension of square matrix only or it is related to square matrix only the only thing that you need to consider here in this case is that whatever elements are there in the leading diagonal they are nonzero rest are all zeros still not understood lets see that means if I have a matrix a and it is denoted as a IJ
the order would be what it would not be M by n it would be rather n by n v because diagonals Merritts is a type of square matrix only and in square matrix the rows and columns are same now a is equal to a IJ n by n is diagonal matrix when when it has the condition that the value is 0 for all I not equal to J that means suppose I have a matrix see the elements are such that it is eight zero zero zero minus one zero zero zero five you see it as a square matrix yes because the number of rows are 3 the number of columns are 3 it is 3 by 3 yes it is a square matrix square matrix okay when will it be the diagonal matrix when all the elements all the elements except the ones in the diagonals diagonals watch the diagonals is the leading diagonal that you need to consider the diagonal is the leading diagonal which is 8 minus 1 5 it is not equal to 0 now no element is 0 the other elements are 0 so it is a diagonal matrix another example of this suppose we have a matrix D and the matrix is 1 0 0
5 a 1 1 is 1 a 1 2 is 0 a 2 1 is 0 and a 2 2 is 5 now in this case what is the order the order is the rows and the columns number of rows are 2 number of columns are also 2 so it is a 2 by 2 matrix and in this 2 by 2 matrix I know that the leading diagonal or the principal diagonal is what it is 1 & 5 the elements 1 & 5 are not equal to 0 the rest elements are 0 so yes this is a diagonal matrix one more thing after the representation and after the examples that you need to consider is suppose I have a diagonals of any order say order n this is of the order 3 this is of the order to that we took right suppose I have a diagonally tricks of not order 3 or 2 or 4 but of any random order say n so how to represent representation is very simple I will show this with the help of example then we will generalize see here it is a 3 by 3 matrix so it is shown as C is equal to diagonal diag is the short form of diagonal what
are the elements in the order a 1 1 is 8 8 to 2 is minus 1 & 8 3 3 is 5 so you can show it in this manner also writing C in the below manner or in the above manner is same right similarly now guess yourself how to write B D is equal to dial in the bracket what you will have the elements of a I AI I wear it I is equal to firstly 1 then 2 then 3 and so on so AI I am means when I is equal to 1 a 1 1 s 1 AI I when I is equal to 2 a 2 2 second row second column is 5 so again writing D in the above manner all the below manner does not change anything it is the same so one thing very important that you need to consider that yes diagonal matrix is possible only for a square matrix the other thing that which diagonals to consider the leading diagonal or the principal diagonal not the other way round so in this video we saw the fourth important type of matrix called as the diagonal matrix in the next video we will be seeing the next important type called us this scalar matrix
12th class Mathematics, Mathematics, CBSE Mathematics, IIT JEE Exam, CBSE Exam, Matrices, Diagonal Matrix
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