Now that the concepts of a permutation and its sign have been defined, the definition of the determinant of a matrix can be given. The use of matrix notation in denoting permutations is merely a matter of convenience. I can reduce to a triangular matrix and now what's the determinant of that triangular matrix? Using (ii) one obtains similar properties of columns. 0000066137 00000 n
Then the determinant of an n × n n \times n n × n matrix … The determinant of A-squared is the determinant of A times the determinant of A. One or minus one, depending whether the number of exchanges was even or the number of exchanges was odd. It's sort of, like, amazing that it can... And the tenth property is equally simple to state, that the determinant of A transposed equals the determinant of A. If a matrix order is n x n, then it is a square matrix. I even wrote here, "plus and minus signs," because this is, like, that's what you have to pay attention to in the formulas and properties of determinants. startxref
And tell me, how do I show that none of this upper stuff makes any difference? Well, then with elimination we know that we can get a row of zeroes, and for a row of zeroes I'm using rule six, the determinant is zero, and that's right. I factor a different two out of the second row, a different two out of the nth row, so I've got all those twos coming out. And then I'll factor out the d2, shall I shall I put the d2 here, and the second row will look like that, and so on. It gives me a combination in row k of the old row and l times this copy of the higher row, and then if -- since I have two equal rows, that's zero, so the determinant after elimination is the same as before. Determinants 4.1 Deﬁnition Using Expansion by Minors Every square matrix A has a number associated to it and called its determinant,denotedbydet(A). We’ll form all n! If a column is all zero, what's the determinant? 0000053613 00000 n
Permutation matrices include the identity matrix and the exchange matrix. OK, so for example, what's the determinant of A inverse? 0000042006 00000 n
basis vector: that is, the matrix is the result of permuting the columns of the identity matrix. » endstream
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So this is rule 3A here. Modify, remix, and reuse (just remember to cite OCW as the source. Courses Why don't -- I'll use an eraser, do it right. So that's great, provided a isn't zero. Mathematics And the transpose is U transpose, l transpose. I just multiply this row by the right number, subtract from that row, kills that. Learn more », © 2001–2018
How about the determinant of l transposed? But I want to see why it's true for n-by-n. 0000065098 00000 n
There's a factor -- this has a factor two in every row, so I can factor two out of the first row. Property two just told us, hey, if we've got two equal rows we. how do I know that the determinant is just a product of That's my proof, really, that once I've got those diagonal entries? We subtract, so I'm putting it in this upper triangular form. In the case of general n the sum is over (n−1)! OK. And again, maybe I won't -- oh, let's see. This is one of over 2,400 courses on OCW. It's not zero, and therefore this makes sense. Oh, because that's still zero zero, right? A permutation matrix is nonsingular, and the determinant is always. Because rule two said that if you do seven row exchanges, then the sign of the determinant reverses. How do you use 3A, which says I can factor out an l, I can factor out a minus l here. Well, one number can't tell you what the whole matrix was. We start from the identity matrix , we perform one interchange and obtain a matrix , we perform a second interchange and obtain another matrix , and so on until at the -th interchange we get the matrix . Row and column expansions. we've got a zero determinant. The determinant of A is then det ( A ) = ε det ( L ) ⋅ det ( U ) . So this is proof, this is proof number ten, using -- well, I don't know which ones I'll use, so I'll put 'em all in, one to nine. Now, exchanging two columns reverses the sign, because I can always, if I want to see why, I can transpose, those columns become rows, I do the exchange, I transpose back. Ahh. What's the determinant of, of A-squared? 0000080910 00000 n
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Finally, this is the point where rule one finally chips in and says that this determinant is one, so it's the product of the d's. Linear Algebra Grinshpan Permutation matrices A permutation matrix is a square matrix obtained from the same size identity matrix by a permutation of rows. 0000004895 00000 n
I have to go back to properties one, two, three. If I double the matrix, what -- so the determinant of A, since I'm writing down, like, facts that follow, the determinant of A-squared is the determinant of A, all squared. So in almost every case, A can factor into LU, and A transposed can factor into that. For instance, associate to the permuta-tion ˙= 24153 the following 5 5 matrix 2 So somehow this proof, this property has to -- somehow the proof of that property -- if we can boil it down to diagonal matrices then we can read it off, whether it's A and A-inverse, or two different diagonal matrices A and B. I'm saying for a diagonal matrices, check. This function is O(n) procedure allocating a buffer of n booleans. which has four rows and four columns. We noted a distinction between two classes of T’s. And so now, actually, what matrices do we now know the determinant of? 315 0 obj <>
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Properties of Determinants The Permutation Expansion is also a convenient starting point for deriving the rule for the determinant of a triangular matrix. What does this tell me about A inverse, its determinant? case. The use of matrix notation in denoting permutations is merely a matter of convenience. So I'll just write them down and use them. For example the matrix rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate That big formula has got too much packed in it. 5. If I -- if I have a box and I double all the sides, I multiply the volume by two to the nth. Suppose, suppose I -- all right, rule seven. �t�Z|pU9Y&�W��&�䨱����Q�H������2ǹ��} �UPPHI �0AAt�5�n ry��im 6+Pd�g�a��,�� ���e��(�+B7����dme��"�W`�@%���c�}�tU�d�X�;\"��lv�5fa����e� ������� if��@����Э�@��\_�c�H � � A�
First, think of the permutation as an operation rather than a list. And all I have to work with is these properties. One or minus one, depending whether the number of exchanges was even or the number of exchanges was odd. That's, like, what I'm headed for but I'm not there yet. If A is a diagonal matrix, then its determinant is just a product of those numbers. Determinant of a triangular matrix. OK, this lecture is like the beginning of the second half of this is to prove. I'm going to keep -- I'm going to have ab cd, but I'm going to subtract l times the first row from the second row. » While the number of transpositions can vary, this So I've factored out all the d's and what I left with? Thanks. Well, I certainly know the determinant of the identity matrix and now I know the determinant of every other matrix that comes from row exchanges from the identities still. Permutation matrices. Actually, you can look ahead to why I need these properties. I'll often write it as D E T A or often also I'll write it as, A with vertical bars, so that's going to mean the determinant of the matrix. 0000081206 00000 n
Where -- where does that come into this rule? But if you do ten row exchanges, the sign of the determinant stays the same, because minus one ten times is plus one. If I had a combination in the second row, then I could use rule two to put it up in the first row, use my property and then use rule two again to put it back, so each row is OK, not only the first row, but each row separately. Such a matrix is always row equivalent to an identity. Now, the second property is what happens if you exchange two rows of a matrix. The sign of the permutation σ is the sign of the determinant of its matrix representation. And it also tells me -- what, just let's, see what else it's telling me. This is also the determinant of the permutation matrix represented by pi. So, like all those properties about rows, exchanging two rows reverses the sign. OK. if I could carry on this board, I could, like, do the two-by-two's. The determinant of a permutation matrix will have to be either 1 or 1 depending on But actually the second property is pretty straightforward too, and then once we get the third we will actually have the determinant. The determinant of a triangular matrix (upper or lower) is given by the product of its diagonal elements. Now, just for a practice, what are all those determinants? Given the LUP [L:low

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