Matrix properties via SVD

rank, nullspace, range

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• rank $r$ of matrix A, is the maximum number of linear independent rows or columns of A (refer to range AND rank)
• in SVD, rank $r$ is also the number of nonzero singular values / nonzero entries on the diagonal of S
• $dimN(A)=n-r$ (refer to fundamental theorem of linear algebra)
• an orthonormal basis spanning N(A): $N(A)=R(V_{nr})$, $V_{nr}=[v_{r+1}\cdots v_n]$
• an orthonormal basis spanning the range of A: $R(A)=R(U_r),U_r=[u_1\cdots u_r]$

matrix norm

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• squared frobenius matrix norm = sum of the diagonal elements of the matrix $A^{T}A$ (trace) = sum of the singular values squared
  • $||A|{|}_F^2=\mathrm{trace}{A}^{T}A={\sum }_{i=1}^n{\sigma }_i^2$
    • ${\sigma }_i^2$: singular values of A
• $I_2$ induced norm $||A|{|}_2$ = largest singular value
  • $||A|{|}_2={\max }_{x\ne 0}\frac{\Vert Ax{\Vert }_2}{\Vert x{\Vert }_2}\equiv {\sigma }_1$