Simplify the Matrix

Problem

[[−1/√(,2),−0.5/2/√(,1.5),1/√(,3)],[1/√(,2),−0.5/2/√(,1.5),1/√(,3)],[0,1/√(,1.5),1/√(,3)]]*[[1,0,0],[0,1,0],[0,0,4]]*[[−1/√(,2),1/√(,2),0],[−0.5/2/√(,1.5),−0.5/2/√(,1.5),1/√(,1.5)],[1/√(,3),1/√(,3),1/√(,3)]]

Solution

1. Simplify the constant terms within the matrices. Note that 1.5=3/2 so √(,1.5)=√(,3/2) The term 0.5/2/√(,1.5) simplifies to 1/2⋅√(,1.5)/2=√(,3)/(4√(,2)) However, observing the structure, let A be the first matrix. The third matrix is AT

2. Identify the components. Let (v_1)=[[−1/√(,2)],[1/√(,2)],[0]] (v_2)=[[−1/√(,6)],[−1/√(,6)],[2/√(,6)]] and (v_3)=[[1/√(,3)],[1/√(,3)],[1/√(,3)]] The simplified middle term −(0.5√(,1.5))/2=−1/4√(,3/2)=−√(,3)/(4√(,2)) does not match standard orthonormal vectors. Re-evaluating the term 1/√(,1.5)=√(,2/3)=2/√(,6) and −0.5/(2/√(,1.5))=−√(,1.5)/4=−√(,6)/8

3. Multiply the first two matrices A*D Since D is diagonal, this scales the columns of A by 1, 1, 4$.

A*D=[[−1/√(,2),−√(,6)/8,4/√(,3)],[1/√(,2),−√(,6)/8,4/√(,3)],[0,2/√(,6),4/√(,3)]]

4. Perform the final matrix multiplication (A*D)*AT

(M_11)=(−1/√(,2))*(−1/√(,2))+(−√(,6)/8)*(−√(,6)/8)+(4/√(,3))*(1/√(,3))=1/2+6/64+4/3=1/2+3/32+4/3=(48+9+128)/96=185/96

(M_12)=(−1/√(,2))*(1/√(,2))+(−√(,6)/8)*(−√(,6)/8)+(4/√(,3))*(1/√(,3))=−1/2+3/32+4/3=(−48+9+128)/96=89/96

(M_13)=(−1/√(,2))*(0)+(−√(,6)/8)*(2/√(,6))+(4/√(,3))*(1/√(,3))=0−1/4+4/3=13/12

(M_22)=(1/√(,2))*(1/√(,2))+(−√(,6)/8)*(−√(,6)/8)+(4/√(,3))*(1/√(,3))=1/2+3/32+4/3=185/96

(M_23)=(1/√(,2))*(0)+(−√(,6)/8)*(2/√(,6))+(4/√(,3))*(1/√(,3))=13/12

(M_33)=(0)*(0)+(2/√(,6))*(2/√(,6))+(4/√(,3))*(1/√(,3))=4/6+4/3=2/3+4/3=2

Final Answer

[[−1/√(,2),−√(,6)/8,1/√(,3)],[1/√(,2),−√(,6)/8,1/√(,3)],[0,2/√(,6),1/√(,3)]]*[[1,0,0],[0,1,0],[0,0,4]]*[[−1/√(,2),1/√(,2),0],[−√(,6)/8,−√(,6)/8,2/√(,6)],[1/√(,3),1/√(,3),1/√(,3)]]=[[185/96,89/96,13/12],[89/96,185/96,13/12],[13/12,13/12,2]]

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