Part III: The Power Stuff: Advanced Techniques in Java

Encode QR Code in Java Part III: The Power Stuff: Advanced Techniques

Add 1 times the first row to the third row and place the result in the third row, like so: J 1 0 0 N J 1N OK O K K 0 1 1O K 5 O OK O K K 0 1 -1O K -1O PL P L Now add 1 times the second row to the third, placing the result in the third row: J1 0 0 N J 1N OK O K K0 1 1O K 5 O OK O K K 0 0 - 2O K - 6 O PL P L Finally, you divide the third row by 2: J 1 0 0 N J 1N OK O K K 0 1 1O K 5 O K OK O K 0 0 1O K 3 O PL P L Note that you now have a triangular matrix on the bottom left; in other words, the bottom left triangle s elements all equal zero This triangle simplifies matters considerably, because this new augmented matrix now represents this system: x=1 y+z=5 z=3 And from the latter two equations, finding the solution for y is pretty trivial: y = 2 Now you ve solved for all three variables Nice work

Did you know that you can name matrices Yup, that s right In matrix terms, a name is denoted in bold For instance, you can name the following matrix, which contains both real and complex elements, A (imaginative, I know): J 1 2 + 2i N O A=K K 3 + 3i 4 O L P

The transpose of a matrix swaps its rows and columns; in other words, the rows become the columns, and the columns become the rows Here s what the transpose of A (called AT ) looks like: J 1 3 + 3i N O AT = K K 2 + 2i 4 O L P You can also define the complex conjugate of a matrix To denote a complex conjugate, you simply add a line above the name of a matrix The complex conjugate of a matrix is simply the complex conjugate element by element of the matrix Here s the complex conjugate of A, where you flip the sign of the imaginary part: J 1 2 - 2i N O A=K K 3 - 3i 4 O L P The adjoint of a matrix is the transpose of the complex conjugate of the matrix You denote an adjoint by adding an asterisk (*) to the name of the matrix Here s what A* looks like: J 1 3 - 3i N O A* = K K 2 - 2i 4 O L P

After you re familiar with the basics of matrices, you can take the next step and start working with a variety of matrix operations, such as addition, subtraction, multiplication, and other fun stuff I explain what you need to know in the following sections

Two matrices are considered equal if every element in the first matrix is equal to the corresponding element in the other matrix For instance, if: J1 2 N O A=K K 3 4O L P And: J1 2 N O B=K K 3 4O L P then A = B Easy enough, right

If you need to, you can add two matrices together Doing so involves adding the elements at corresponding positions in the two matrices For example, if: J1 2 N J5 6 N O K O A=K K 3 4 O and B = K 7 8 O L P L P then: J1 2 N J5 6 N J 6 8 N O K O K O A+B=K K 3 4 O + K 7 8 O = K 10 12O L P L P L P And just so you know: A + B = B + A