Mathematics & Numerical Analysis

Matrix representation of 2D clockwise rotation:

• $(0, 1)$ is obtained by rotating $(1, 0)$ counter-clockwise by $90^\circ$.
• Length must be preserved ($1$), but $(1, -1)$ has norm $\sqrt{2}$.
• clockwise rotation by $90^\circ$ corresponds to an angle $\theta = -90^\circ$. The transformation matrix is: $\begin{pmatrix} \cos(-90^\circ) & -\sin(-90^\circ) \\ \sin(-90^\circ) & \cos(-90^\circ) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$. Applying this to the standard basis vector $(1, 0)^T$: $T(1, 0) = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ -1 \end{pmatrix} = (0, -1)$.
• $(1, 1)$ does not preserve length.

Eigenvalues of an involutory matrix ($A^2 = I$):

• $0, 1$ are eigenvalues of idempotent matrices ($A^2 = A$).
• If $\lambda$ is an eigenvalue of $A$ with eigenvector $v \ne 0$, then $A v = \lambda v$. Premultiplying by $A$: $A^2 v = \lambda A v = \lambda^2 v$. Since $A^2 = I$, we have $I v = v = \lambda^2 v \implies \lambda^2 = 1 \implies \lambda = \pm 1$. Thus, the only possible eigenvalues of an involutory matrix are $+1$ and $-1$.
• Restricts to positive eigenvalues only.
• Restricts to negative eigenvalues only.

Properties of real skew-symmetric matrices:

• By definition of a skew-symmetric matrix, $A^T = -A$. Taking determinants: $\det(A^T) = \det(-A) = (-1)^n \det(A)$. Since $\det(A^T) = \det(A)$ and $n = 5$ (odd): $\det(A) = -\det(A) \implies 2\det(A) = 0 \implies \det(A) = 0$. Since the determinant is zero, $A$ is definitely a singular matrix.
• An invertible matrix must have a non-zero determinant.
• real skew-symmetric matrix of odd order always has at least one real eigenvalue equal to zero ($\lambda = 0$).
• The determinant is identically zero.

Linear independence axiomatic properties:

• singleton set containing one non-zero vector $\{v\}$ is always linearly independent because $c v = 0 \implies c = 0$.
• The empty set is independent by definition, not dependent.
• In linear algebra, the empty set $\emptyset$ is defined as linearly independent vacuously, because it contains no vectors that can form a non-trivial linear combination equaling the zero vector.
• Sets containing the zero vector or collinear vectors are linearly dependent.