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| The test for orthogonality of two vectors is '''''X'''^T^'''Y''' = 0''. | The test for orthogonality of two vectors is ''x^T^y = 0''. |
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| For a vector '''''X''''', the total length can be thought of as the sum of each components' absolute value. If '''''X''''' is ''[1 2 3]'', the length is 6. The squared length can be thought of as the sum of each components' square. For the same '''''X''''', this is 14. This can be generalized as '''''X'''^T^'''X'''''. | For a vector ''x'', the total length can be thought of as the sum of each components' absolute value. If ''x'' is ''[1 2 3]'', the length is 6. The squared length can be thought of as the sum of each components' square. For the same ''x'', this is 14. This can be generalized as ''x^T^x''. |
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| For a similar reason, the total length of the hypotenuse can be thought of as the sum of the other two vectors. Instead of characterizing a vector '''''Z''''', we can use '''''(X+Y)^T^(X+Y)'''''. For example, if '''''X''''' is ''[1 2 3]'' and '''''Y''''' is ''[2 -1 0]'', it should be understood that '''''Z''''' is ''[3 1 3]''. | For a similar reason, the total length of the hypotenuse can be thought of as the sum of the other two vectors. Instead of characterizing a vector ''z'', we can use ''(x+y)^T^(x+y)''. For example, if ''x'' is ''[1 2 3]'' and ''y'' is ''[2 -1 0]'', it should be understood that ''z'' is ''[3 1 3]''. |
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| If the vectors '''''X''''' and '''''Y''''' are perpendicular, or '''orthogonal''', then the Pythagorean theorem should hold. '''''X'''^T^'''X''' + '''Y'''^T^'''Y''' = ('''X'''+'''Y''')^T^('''X'''+'''Y''')''. This expands to '''''X'''^T^'''X''' + '''Y'''^T^'''Y''' = '''X'''^T^'''X''' + '''Y'''^T^'''Y''' + '''X'''^T^'''Y''' + '''Y'''^T^'''X'''''. By cancelling out common terms, this simplifies to ''0 = '''X'''^T^'''Y''' + '''Y'''^T^'''X'''''. | If the vectors ''x'' and ''y'' are perpendicular, or '''orthogonal''', then the Pythagorean theorem should hold. ''x^T^x + y^T^y = (x+y)^T^(x+y)''. This expands to ''x^T^x + y^T^y = x^T^x + y^T^y + x^T^y + y^T^x''. By cancelling out common terms, this simplifies to ''0 = x^T^y + y^T^x''. |
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| It must be understood that the last two terms are the same value. Therefore, this further simplifies to ''0 = 2'''X'''^T^'''Y''''' and finally to ''0 = '''X'''^T^'''Y'''''. | It must be understood that the last two terms are the same value. Therefore, this further simplifies to ''0 = 2x^T^y'' and finally to ''0 = x^T^y''. |
Orthogonality
Contents
Test
The test for orthogonality of two vectors is xTy = 0.
The Pythagorean theorem specifies that for two sides of a right triangle, x and y, the hypotenuse z is characterized by x2 + y2 = z2.
For a vector x, the total length can be thought of as the sum of each components' absolute value. If x is [1 2 3], the length is 6. The squared length can be thought of as the sum of each components' square. For the same x, this is 14. This can be generalized as xTx.
For a similar reason, the total length of the hypotenuse can be thought of as the sum of the other two vectors. Instead of characterizing a vector z, we can use (x+y)T(x+y). For example, if x is [1 2 3] and y is [2 -1 0], it should be understood that z is [3 1 3].
If the vectors x and y are perpendicular, or orthogonal, then the Pythagorean theorem should hold. xTx + yTy = (x+y)T(x+y). This expands to xTx + yTy = xTx + yTy + xTy + yTx. By cancelling out common terms, this simplifies to 0 = xTy + yTx.
It must be understood that the last two terms are the same value. Therefore, this further simplifies to 0 = 2xTy and finally to 0 = xTy.
Application
For a subspace S to be orthogonal to a subspace T, every vector in S must be orthogonal to every vector in T.
The null space of A (a.k.a. N(A)) is orthogonal to the row space of A (a.k.a. R(A)). The null space of AT (a.k.a. N(AT)) is orthogonal to the column space of A (a.k.a. C(A)).