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Vectors are said to be orthogonal if their dot product is zero.Vectors in Rn are perpendicular if they are nonzero and orthogonal.
If the product of two matrices is the identity matrix then one matrix is the inverse or reciprocal of the other matrix. EXAMPLE A =(4 1) A-1 = (0.3 -0.1) then AA-1 = (1 0) .....(2 3)......... (-0.2 0.4)................... (1 1) The dots simply maintain the spacing and serve no other purpose.
No, it is not.
All vectors that are perpendicular (their dot product is zero) are orthogonal vectors.Orthonormal vectors are orthogonal unit vectors. Vectors are only orthonormal if they are both perpendicular have have a length of 1.
yes it is
First let's be clear on the definitions.A matrix M is orthogonal if MT=M-1Or multiply both sides by M and you have1) M MT=Ior2) MTM=IWhere I is the identity matrix.So our definition tells us a matrix is orthogonal if its transpose equals its inverse or if the product ( left or right) of the the matrix and its transpose is the identity.Now we want to show why the inverse of an orthogonal matrix is also orthogonal.Let A be orthogonal. We are assuming it is square since it has an inverse.Now we want to show that A-1 is orthogonal.We need to show that the inverse is equal to the transpose.Since A is orthogonal, A=ATLet's multiply both sides by A-1A-1 A= A-1 ATOr A-1 AT =ICompare this to the definition above in 1) (M MT=I)do you see how A-1 now fits the definition of orthogonal?Or course we could have multiplied on the left and then we would have arrived at 2) above.
Vectors are said to be orthogonal if their dot product is zero.Vectors in Rn are perpendicular if they are nonzero and orthogonal.
no
If the product of two matrices is the identity matrix then one matrix is the inverse or reciprocal of the other matrix. EXAMPLE A =(4 1) A-1 = (0.3 -0.1) then AA-1 = (1 0) .....(2 3)......... (-0.2 0.4)................... (1 1) The dots simply maintain the spacing and serve no other purpose.
If the multiplicative inverse exists then, by definition, the product is 1 which is rational.
No, it is not.
Three of them are "orthogonal", "orthodontist", and "orthopedic", and "orthogonal" is a very important word in mathematics. For one example, two vectors are orthogonal whenever their dot product is zero. "Orthogonal" also comes into play in calculus, such as in Fourier Series.
All vectors that are perpendicular (their dot product is zero) are orthogonal vectors.Orthonormal vectors are orthogonal unit vectors. Vectors are only orthonormal if they are both perpendicular have have a length of 1.
yes it is
You can indicate the multiplication with a multiplication sign. If your matrices are "A" and "B", the product is: A x B In other words, you are indicating the product, but not actually carrying out any multiplication. Anybody who understands about matrices should know what this refers to.
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If the product of two matrices is an identity matrix then, one matrix is inverse of the other. i.e. AB = I then, A = B-1 and B = A-1Inverse of matrix can be found by using these two results:A = AI and A = IA.By using these results inverse of a matrix can be found by applying same elementary row or column operation on both sides. A on R.H.S. remains as it is.