public void toReducedRowEchelonForm() {
InvertibleBinaryOperation<T> addition = ring.addition();
InvertibleBinaryOperation<T> multiplication =
(InvertibleBinaryOperation<T>) ring.multiplication();
toRowEchelonForm();
int j = 0;
for (int i = 0; i < m; i++) {
while (j < n) {
T aij = get(i, j);
if (!aij.equals(zero)) {
aij = multiplication.inverse(aij);
for (int k = j + 1; k < n; k++) {
T aik = multiplication.op(get(i, k), aij);
set(i, k, aik);
for (int l = 0; l < i; l++)
set(
l,
k,
addition.op(get(l, k), addition.inverse(multiplication.op(aik, get(l, j)))));
}
set(i, j, one);
for (int l = 0; l < i; l++) set(l, j, zero);
j++;
break;
}
j++;
}
}
}
public Matrix<T> getKernelBasis() {
Matrix<T> reducedRowEchelonForm = getReducedRowEchelonForm();
boolean[] hasLeadingOne = new boolean[n];
List<Integer> indices = new ArrayList<Integer>();
int j = 0;
for (int i = 0; i < m; i++)
while (j < n) {
T aij = reducedRowEchelonForm.get(i, j);
if (!aij.equals(zero)) {
if (!aij.equals(one)) throw new Error();
indices.add(j);
hasLeadingOne[j] = true;
j++;
break;
}
j++;
}
InvertibleBinaryOperation<T> addition = ring.addition();
Matrix<T> kernelBasis = new Matrix<T>(ring, n, n - indices.size());
kernelBasis.zero();
int k = 0;
for (int i = 0; i < n; i++)
if (!hasLeadingOne[i]) {
kernelBasis.set(i, k, one);
for (int l = 0; l < indices.size(); l++)
kernelBasis.set(indices.get(l), k, addition.inverse(reducedRowEchelonForm.get(l, i)));
k++;
}
if (k != kernelBasis.n) throw new Error(k + " / " + indices.size());
return kernelBasis;
}
public void toRowEchelonForm() {
InvertibleBinaryOperation<T> plus = ring.addition();
BinaryOperationWithIdentity<T> times = ring.multiplication();
for (int row = 0, column = 0; column < n; column++) {
int pivot;
for (pivot = row; pivot < m; pivot++) if (!get(pivot, column).equals(zero)) break;
if (pivot != m) { // is this column already done?
if (pivot != row)
// swap rows row and pivot
for (int j = column; j < n; j++) {
T tmp = get(pivot, j);
set(pivot, j, get(row, j));
set(row, j, tmp);
}
for (int i = row + 1; i < m; i++)
if (!(get(i, column).equals(zero))) {
for (int j = column + 1; j < n; j++)
// a_i,j = a_row,column * a_i,j - a_row,j * a_i,column
set(
i,
j,
plus.op(
times.op(get(row, column), get(i, j)),
plus.inverse(times.op(get(row, j), get(i, column)))));
set(i, column, zero);
}
row++;
}
}
}
public static <T> Matrix<T> subspaceIntersection(Matrix<T> m1, Matrix<T> m2) {
if (m1.m != m2.m) throw new IllegalArgumentException();
if (m1.ring != m2.ring) throw new IllegalArgumentException();
Matrix<T> m = new Matrix<T>(m1.ring, m1.m, m1.n + m2.n);
for (int i = 0; i < m1.m; i++) {
for (int j = 0; j < m1.n; j++) m.set(i, j, m1.get(i, j));
for (int j = 0; j < m2.n; j++) m.set(i, m1.n + j, m2.get(i, j));
}
Matrix<T> kernelBasis = m.getKernelBasis();
return product(m1, kernelBasis.getSubmatrix(m1.n, kernelBasis.n));
}
public void solveFor(Matrix<T> b) {
if (m != n || m != b.m) throw new IllegalArgumentException();
Field<T> field = (Field<T>) ring;
InvertibleBinaryOperation<T> times = field.multiplication();
InvertibleBinaryOperation<T> plus = field.addition();
for (int i = 0; i < m; i++) {
int pivotIndex = i;
for (; pivotIndex < m; pivotIndex++) if (!get(pivotIndex, i).equals(zero)) break;
if (pivotIndex == m) throw new ArithmeticException("singular system");
// swap two equations to bring the pivot to the diagonal
Object[] swapRow = matrix[pivotIndex];
matrix[pivotIndex] = matrix[i];
matrix[i] = swapRow;
for (int j = 0; j < b.n; j++) {
T swap = b.get(pivotIndex, j);
b.set(pivotIndex, j, b.get(i, j));
b.set(i, j, swap);
}
// normalize current row.
T pivot = times.inverse(get(i, i));
for (int j = i + 1; j < n; j++) set(i, j, times.op(get(i, j), pivot));
for (int j = 0; j < b.n; j++) b.set(i, j, times.op(b.get(i, j), pivot));
// subtract multiples of it from the ones below.
for (int k = i + 1; k < m; k++) {
T first = get(k, i);
if (!first.equals(zero)) {
for (int j = i + 1; j < n; j++)
// buf [k] [j] -= first * buf [i] [j];
set(k, j, plus.op(get(k, j), plus.inverse(times.op(first, get(i, j)))));
for (int j = 0; j < b.n; j++)
// x [k] -= first * x [i];
b.set(k, j, plus.op(b.get(k, j), plus.inverse(times.op(first, b.get(i, j)))));
}
}
}
// now back-substitute
for (int k = m - 1; k > 0; k--)
for (int i = 0; i < k; i++)
for (int j = 0; j < b.n; j++)
b.set(i, j, plus.op(b.get(i, j), plus.inverse(times.op(get(i, k), b.get(k, j)))));
}
private T determinantByDimensionalReduction(int k) {
if (k == 0) return get(0, 0);
InvertibleBinaryOperation<T> plus = ring.addition();
BinaryOperationWithIdentity<T> times = ring.multiplication();
T result =
times.op(
plus.op(get(k, k), plus.inverse(times.identity())),
determinantByDimensionalReduction(k - 1));
for (int i = 0; i < k; i++)
for (int j = 0; j < k; j++)
set(i, j, plus.op(get(i, j), plus.inverse(times.op(get(i, k), get(k, j)))));
result = plus.op(result, determinantByDimensionalReduction(k - 1));
for (int i = 0; i < k; i++)
for (int j = 0; j < k; j++) set(i, j, plus.op(get(i, j), times.op(get(i, k), get(k, j))));
return result;
}
public Matrix<T> getSubmatrix(int m1, int m2, int n1, int n2) {
if (m1 > m2 || n1 > n2 || m1 < 0 || n2 < 0 || m2 > m || n2 > n)
throw new IllegalArgumentException(
"[" + m1 + "," + m2 + "] x [" + n1 + "," + n2 + "] from " + m + " x " + n);
Matrix<T> submatrix = new Matrix<T>(ring, m2 - m1, n2 - n1);
for (int i = m1; i < m2; i++)
for (int j = n1; j < n2; j++) submatrix.set(i - m1, j - n1, get(i, j));
return submatrix;
}
public static <T> Matrix<T> sum(Matrix<T> A, Matrix<T> B) {
if (A.ring != B.ring || A.m != B.m || A.n != B.n)
throw new IllegalArgumentException(A.m + " x " + A.n + ", " + B.m + " x " + B.n);
InvertibleBinaryOperation<T> addition = A.ring.addition();
Matrix<T> sum = new Matrix<T>(A.ring, A.m, A.n);
for (int i = 0; i < A.m; i++)
for (int j = 0; j < A.n; j++) sum.set(i, j, addition.op(A.get(i, j), B.get(i, j)));
return sum;
}
public static <T> Matrix<T> product(Matrix<T> A, Matrix<T> B) {
if (A.ring != B.ring || A.n != B.m) throw new IllegalArgumentException();
BinaryOperation<T> addition = A.ring.addition();
BinaryOperation<T> times = A.ring.multiplication();
Matrix<T> product = new Matrix<T>(A.ring, A.m, B.n);
for (int i = 0; i < A.m; i++)
for (int k = 0; k < B.n; k++) {
T t = A.zero;
for (int j = 0; j < A.n; j++) t = addition.op(t, times.op(A.get(i, j), B.get(j, k)));
product.set(i, k, t);
}
return product;
}
public void add(int i, int j, T t) {
set(i, j, ring.addition().op(get(i, j), t));
}
public Matrix<T> changeRing(Ring<T> ring) {
Matrix<T> result = new Matrix<T>(ring, m, n);
for (int i = 0; i < m; i++) for (int j = 0; j < n; j++) result.set(i, j, get(i, j));
return result;
}
public Matrix<T> getTranspose() {
Matrix<T> transpose = new Matrix<T>(ring, n, m);
for (int i = 0; i < m; i++) for (int j = 0; j < n; j++) transpose.set(j, i, get(i, j));
return transpose;
}