A matrix with square diagonals and transpose being increments of other

Construct a matrix with the following property:

1. North-west to South East diagonals are squares.

2. Matrix[i][j] + 1 = Matrix[j][i] for each i less than j.

Example of such m a matrix is here. Also, since it appears this matrix does not have a name, I have named my class "Sqaure Diagonal, Transpose Increment".

Looking for code review, best practices, clever optimzations etc.

public final class SquareDiagonalTransposeIncrement {

private SquareDiagonalTransposeIncrement() {}

public static int[][] getSqaureDiagonalTransposeIncrementMatrix (int dimension) {
if (dimension < 1) {
throw new IllegalArgumentException("dimension: " + dimension + " should be greater than 0");
}

final int[][] m = new int[dimension][dimension];

// set the diagonal
for (int i = 0; i < dimension; i++) {
m[i][i] = (i + 1) * (i + 1);
}

/*
* Definition of the quadrant.
*
* - 0th quadrant means a quadrant square matrix from (0,0) to (0,0)
* - 1st quadrant means a quadrant square matrix from (0,0) to (1,1)
* - 2nd quadrant means a quadrant square matrix from (0,0) to (2,2)
*
*/
for (int quadrant = 0; quadrant < dimension - 1; quadrant++) {
}

return m;
}

private static void besiegeQuadrant(int[][] m, int quadrant) {

// besiege the quadrant from the eastern and southern boundaries.
for (int i = 0; i <= quadrant; i++) {
m[i][quadrant + 1] = ++value; // eastern
m[quadrant + 1][i] = ++value; // southern
}
}

public static void main(String[] args) {
int[][] m =  SquareDiagonalTransposeIncrement.getSqaureDiagonalTransposeIncrementMatrix (4);
assertArrayEquals(new int[]{ 1, 2, 5, 10}, m);
assertArrayEquals(new int[]{ 3, 4, 7, 12}, m);
assertArrayEquals(new int[]{ 6, 8, 9, 14}, m);
assertArrayEquals(new int[]{11,13,15, 16}, m);
}
}

Throwing a different spin on the process, I think you are missing an alternate algorithm that produces the same result....

Here is a 'simple' loop that 'works'. It works because it uses a different 'rule' that is based on the pattern that your matrix creates.

The Pattern is obvious when you study the matrix from the bottom-right corner, and work backwards. The initial value is the square of the matrix size. Then you subtract 1 to the left, then another 1 above, and another 1 to the next left, and another 1 to the next above, etc.

private static final int[][] buildMagicMatrix(int size) {
int value = size * size + 1;
int[][] matrix = new int[size][size];
for (int i = size - 1; i >= 0; i--) {
matrix[i][i] = --value;
for (int d = i - 1; d >= 0; d--) {
matrix[i][d] = --value;
matrix[d][i] = --value;
}
}
return matrix;
}

Using this algorithm you produce results like:

Size 9
1    2    5   10   17   26   37   50   65
3    4    7   12   19   28   39   52   67
6    8    9   14   21   30   41   54   69
11   13   15   16   23   32   43   56   71
18   20   22   24   25   34   45   58   73
27   29   31   33   35   36   47   60   75
38   40   42   44   46   48   49   62   77
51   53   55   57   59   61   63   64   79
66   68   70   72   74   76   78   80   81

I don't believe that your description uniquely defines the matrix. However, you can fill the upper-right triangle merely by pattern recognition. Starting from the diagonal going up, each column is a sequence descending by 2. From there, just apply the transpose-increment rule.

public static int[][] getSqaureDiagonalTransposeIncrementMatrix(int dim) {
int[][] m = new int[dim][dim];

// Fill diagonal and upper-right
for (int row = 0, col = row; row < dim; row = ++col) {
m[row][col] = (row + 1) * (row + 1);
for (--row; row >= 0; --row) {
m[row][col] = m[row + 1][col] - 2;
}
}
// Fill the lower-left using the transpose-increment rule
for (int row = 0; row < dim; ++row) {
for (int col = 0; col < row; ++col) {
m[row][col] = m[col][row] + 1;
}
}
return m;
}