for any point rn the point in that is closest to is its orthogonal projection pxas the following theorem shows theorem orthogonal projection and minimal distance let { uk be an orthonormal basis of subspace and let be the orthogonal projection matrix onto the solution to the minimization program min kx yk yv is px that ispx is closest to proofwe can write each point as kx yk = ai ui = pk = ai ui consequently ai ui kxk = ai hxui = minimizing this with respect to the {ai gives ai hxui ii in view of ( )the optimal is thus px eigenvalues and eigenvectors let be an matrix if av lv for some number and non-zero vector vthen is called an eigenvalue of with eigenvector if (lvis an (eigenvalueeigenvectorpairthe matrix li maps any multiple of to the zero vector consequentlythe columns of li are linearly dependentand hence its determinant is this provides way to identify the eigenvaluesnamely as the different roots lr of the characteristic polynomial eigenvalue eigenvector characteristic polynomial det(li ( ) ( lr )ar where ar the integer ai is called the algebraic multiplicity of li the eigenvectors that correspond to an eigenvalue li lie in the kernel or null space of the matrix li athat isthe linear space of vectors such that (li ) this space is called the eigenspace of li its dimensiondi { }is called the geometric multiplicity of li it always holds that di ai if di nthen we can construct basis for rn consisting of eigenvectorsas illustrated next example (linear transformation (cont )we revisit the linear transformation in figure where - / - the characteristic polynomial is ( )( / with roots - / / - and - / / the corresponding unit eigenvectors are [ - ]and [ - ]the eigenspace corresponding to is algebraic multiplicity null space geometric multiplicity