Question

If A and B are arbitrary real m x n matrices, then the mapping u27e8A, Bu27e9 = trace(A^T B) defines an inner product in u211d^{m u00d7 n}. Use this inner product to find u27e8A, Bu27e9, the norms ||A|| and ||B||, and the angle (in radians) u03b1_{A,B} between A and B for A = egin{bmatrix} 3 & -1 \ -2 & 3 \ -2 & -2 end{bmatrix} and B = egin{bmatrix} -1 & -1 \ -3 & 3 \ 2 & -1 end{bmatrix}. u27e8A, Bu27e9 = 11 ||A|| = sqrt31 ||B|| = 5 u03b1_{A,B} = 66.726 

          If A and B are arbitrary real m x n matrices, then the mapping 
u27e8A, Bu27e9 = trace(A^T B)
defines an inner product in u211d^{m u00d7 n}. Use this inner product to find u27e8A, Bu27e9, the norms ||A|| and ||B||, and the angle (in radians) u03b1_{A,B} between A and B for 
A = egin{bmatrix} 3 & -1 \ -2 & 3 \ -2 & -2 end{bmatrix} and B = egin{bmatrix} -1 & -1 \ -3 & 3 \ 2 & -1 end{bmatrix}. 
u27e8A, Bu27e9 = 11
||A|| = sqrt31
||B|| = 5
u03b1_{A,B} = 66.726
Show more…
If A and B are arbitrary real m x n matrices, then the mapping u27e8A, Bu27e9 = trace(A^T B) defines an inner product in u211d^m u00d7 n. Use this inner product to find u27e8A, Bu27e9, the norms ||A|| and ||B||, and the angle (in radians) u03b1A,B between A and B for A = eginbmatrix 3 -1 -2 3 -2 -2 endbmatrix and B = eginbmatrix -1 -1 -3 3 2 -1 endbmatrix. u27e8A, Bu27e9 = 11 ||A|| = sqrt31 ||B|| = 5 u03b1A,B = 66.726

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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If A and B are arbitrary real m x n matrices, then the mapping (A,B) = trace(ATB) defines an inner product in Rmxn. Use this inner product to find (A, B), the norms ||A|| and ||B||, and the angle (in radians) α between A and B for A and B.
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Transcript

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00:01 Okay, so in this question we're given two matrices, matrices a and b of the same type, we define an inner product on the space of these matrices, which is a trace of the transpose of a times b, and we're asked to find the product, the norms, and the angle.
00:15 So let's start with the inner product, right? so the first thing we'll have to do is calculate a transpose times b.
00:21 So transpose of a will of course be 3, minus 2, minus 2, minus 1, 3, minus 2, and this multiplies by minus 1, minus 1, minus 3, 3, 2, minus 1.
00:35 So in the end, we're going to, so this has two rows, this has two columns.
00:39 So the final results will be 2 by 2.
00:42 And the first entry then will be first row times first column.
00:46 So minus 3, plus 6 is 3, minus 4 is minus 1.
00:52 Second column now, so minus 3, minus 6 is minus 9.
01:00 Plus 2 is minus 7.
01:03 Then second row will be minus 1 times minus 1 is plus 1.
01:07 Minus 9 is minus 8, minus 4 is minus 12.
01:13 And then plus 1 plus 6, plus 9, so 10, plus 2 is 12.
01:21 Right.
01:22 So from here we conclude that the inner product between a and b, which is a trace of the matrix above, 8 transpose times b, is the sum of the diagonal.
01:31 Is minus 1 plus 12, which is 11.
01:36 Okay.
01:39 Now, second thing, we want the norm of a, and the norm of a, as you know, is a square root of a against a, right? so the first thing that we should do is calculate a transpose against a, which will be, so 3 minus 2, minus 2, minus 1, 3 minus 2, and he'll be 3 minus 2, minus 2, minus 1, 3, minus 2.
02:07 So let's see what we get.
02:12 So these will be 9, 3 squares, so 9, plus 4 is 13, plus 4 is 17.
02:21 That the other one will be minus 3, minus 6 is minus 9, plus 4 is minus 5.
02:30 Then here we'll get minus 5 again, but let's just double checks.
02:32 Is in minus 3, minus 6 is minus 9, plus 4 is minus 5, and then the other one will be 1 plus 9 is 10, plus 4 is 14.
02:43 All right.
02:45 From where we conclude, let's just finish in here.
02:47 So we'll be the square root of 17 plus 14 is a trace.
02:51 So 17 plus 4 is 21, plus 10 is 31.
02:54 Right? so this is the square root of 31.
02:59 Correct? i'm just checking that we get the same answers as you put in in the question.
03:03 Now let's go for the norm of b.
03:07 This will be square root of b against b.
03:14 So let's make the calculation in here.
03:16 Now b transpose times b.
03:18 So this will be what was b.
03:20 So minus 1, minus 3, 2.
03:25 Second column here was minus 1, 3 minus 1...
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