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Home > Gaussian elimination - LU decomposition - Vector spaces associated with Matrices - Special ortho

Gaussian elimination - LU decomposition - Vector spaces associated with Matrices - Special ortho

Identify the augmented matrix from the system of linear equations.

Perform row operations to convert the matrix to upper triangular form.

Swap rows if necessary to position a non-zero pivot.

Scale rows to make the pivot equal to 1.

Eliminate entries below the pivot by adding or subtracting multiples of the pivot row.

Back substitute to find the solution to the system.

Start from the last row and substitute upwards to solve for each variable.

Start with the matrix you want to decompose.

Initialize matrices L (lower triangular) and U (upper triangular).

Perform Gaussian elimination without row exchanges to fill in U.

Fill in L by recording multipliers used during elimination.

Ensure that the diagonal of L consists of 1s.

Verify the decomposition by checking that \( A = LU \).

Identify the matrix and its dimensions.

Determine the column space (C(A))

Find the span of the columns of the matrix.

Determine the null space (N(A))

Solve the homogeneous equation \( Ax = 0 \) to find the solutions.

Determine the row space (R(A))

Find the span of the rows (equivalent to the column space of the transpose).

Identify the rank and nullity of the matrix

Rank is the dimension of the row space (or column space).

Nullity is the dimension of the null space.

Define a special orthogonal matrix \( Q \)

Ensure \( Q^T Q = I \) (orthogonality).

Check if \( \det(Q) = 1 \) (special property).

Verify that the columns of \( Q \) are orthonormal

Each column vector has a norm of 1.

The dot product between distinct column vectors is 0.

Consider applications of special orthogonal matrices in rotations and reflections in Euclidean space.

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