Find the Norm [[1,1,1,0],[1,-2y,-4z,6],[-x,-y,-3z,6]]

Problem

‖[1,1,1,0],[1,−2*y,−4*z,6],[−x,−y,−3*z,6]‖F

Solution

1. Identify the type of norm to be calculated. For a non-square matrix, the Frobenius norm is the standard norm, defined as the square root of the sum of the squares of all its entries.

2. Square each individual entry of the matrix.

1=1

1=1

1=1

0=0

1=1

(−2*y)2=4*y2

(−4*z)2=16*z2

6=36

(−x)2=x2

(−y)2=y2

(−3*z)2=9*z2

6=36

3. Sum all the squared entries together.

S=1+1+1+0+1+4*y2+16*z2+36+x2+y2+9*z2+36

4. Combine like terms to simplify the expression under the radical.

S=x2+(4*y2+y2)+(16*z2+9*z2)+(1+1+1+1+36+36)

S=x2+5*y2+25*z2+76

5. Apply the square root to the sum to find the Frobenius norm.

‖A‖F=√(,x2+5*y2+25*z2+76)

Final Answer

‖[1,1,1,0],[1,−2*y,−4*z,6],[−x,−y,−3*z,6]‖F=√(,x2+5*y2+25*z2+76)

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