Find the Norm

Problem

[[4,1,−2,3],[1,1/4,−1/2,3/4],[2,1/2,−1,1]]

Solution

1. Identify the matrix A and observe that the rows are linearly dependent. Specifically, the second row is 1/4 times the first row, and the third row is 1/2 times the first row.

2. Choose the Frobenius norm (∥A∥ as the standard norm for a matrix, which is the square root of the sum of the squares of all elements.

3. Calculate the sum of the squares of the elements in the first row:

4+1+(−2)2+3=16+1+4+9=30

4. Calculate the sum of the squares of the elements in the second row:

1+(1/4)2+(−1/2)2+(3/4)2=1+1/16+1/4+9/16=(16+1+4+9)/16=30/16=15/8

5. Calculate the sum of the squares of the elements in the third row:

2+(1/2)2+(−1)2+1=4+1/4+1+1=6+1/4=25/4

6. Sum the results from all rows:

30+15/8+25/4=(240+15+50)/8=305/8

7. Take the square root to find the Frobenius norm:

∥A∥=√(,305/8)=√(,305)/(2√(,2))=√(,610)/4

Final Answer

∥[[4,1,−2,3],[1,1/4,−1/2,3/4],[2,1/2,−1,1]]∥=√(,610)/4

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