Find the Norm

Problem

‖[5,1,2,8,4,3,8,1],[5,9,1,5,7,2,9,1],[7,6,1,8,3,1,8,2],[0,5,0,8,5,4,6,7],[2,8,6,4,1,5,2,1],[0,0,0,0,0,0,0,0]‖_F

Solution

1. Identify the type of norm to be calculated. For a matrix, the default norm is typically the Frobenius norm, which is the square root of the sum of the squares of all elements.

2. Square each individual element in the matrix.

5^2,1^2,2^2,8^2,4^2,3^2,8^2,1^2⇒25,1,4,64,16,9,64,1

5^2,9^2,1^2,5^2,7^2,2^2,9^2,1^2⇒25,81,1,25,49,4,81,1

7^2,6^2,1^2,8^2,3^2,1^2,8^2,2^2⇒49,36,1,64,9,1,64,4

0^2,5^2,0^2,8^2,5^2,4^2,6^2,7^2⇒0,25,0,64,25,16,36,49

2^2,8^2,6^2,4^2,1^2,5^2,2^2,1^2⇒4,64,36,16,1,25,4,1

3. Sum the squared values for each row.

Row 1 sum=25+1+4+64+16+9+64+1=184

Row 2 sum=25+81+1+25+49+4+81+1=267

Row 3 sum=49+36+1+64+9+1+64+4=228

Row 4 sum=0+25+0+64+25+16+36+49=215

Row 5 sum=4+64+36+16+1+25+4+1=151

Row 6 sum=0

4. Calculate the total sum of all squared elements.

Total Sum=184+267+228+215+151+0=1045

5. Take the square root of the total sum to find the Frobenius norm.

√(,1045)≈32.32646

Final Answer

‖[5,1,2,8,4,3,8,1],[5,9,1,5,7,2,9,1],[7,6,1,8,3,1,8,2],[0,5,0,8,5,4,6,7],[2,8,6,4,1,5,2,1],[0,0,0,0,0,0,0,0]‖_F=√(,1045)

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