Linear Algebra Observations

ker(A) = ker(A^tA)

x is a vector.

The kernel of A is all x such that Ax = 0.

\lim_{n \rightarrow \infty} z_n

A^tAx = A^t(Ax).

So for all x such that Ax is 0, A^t(Ax) must also be 0. Therefore, the kernel of A is also in A^tAx.

Alternative characterizaton of kernel.

Kernel is the part of a vector that “doesn’t count” in a transformation.

Invertible and linearly independent variables can also be characterized in terms of whether multiple x can correlate to a single x0.

Arbitratry solution x in Ax=b can be written as x = xh+x0. Where xh is part of kerA and x0 is part of perp(kerA). Obviously Ax = Axh+Ax0 and Axh = 0. So Ax0 = b and Ax = b. Ax0 = Ax. For all x, Ax = Ax0 where x0 is the component of the vector x within perp(kerA). If both x0 and x1 are in perp(kerA),

Kernel is also the eigenspace with an eigenvalue of 0.

Orthogonoal and Unitary Matrices

Unitary matricies are those whose adjoint, or transpose conjugate, is its inverse, thus U*U-dagger = I. Orthogonal matrices are simply unitary matrices that are composed of all real members. Orthogonal transformations preserve vector length.