Important Linear Algebra Properties

1. $\mathrm{Rank}(A)=\mathrm{Rank}(A^{T}A)$.

   • Prove by showing $N(A)=N(A^{T}A)$ and rank-nullity theorem

2. $A^{T}A+\lambda I$ is invertible for all $\lambda \ge 0$, no matter the rank of $A$.

3. Spectral theorem says that, if $A$ is a real symmetric matrix,

   • Eigenvalues of $A$ are real.
   • Eigenvectors corresponding to different eigenvalues are linearly independent.
   • $A$ is orthogonally diagonalizable.
   • Symmetric matrices always have enough linearly independent eigenvectors to diagonalize the matrix.

4. Eigenvectors corresponding to distinct eigenvalues are always linearly independent.

5. $1\le \text{geometric multiplicity}\le \text{arithmetic multiplicity}$ for all eigenvalues of a matrix.

   • If some eigenvalue has its AM $>1$ but AM $=$ GM, then it has AM number of linearly independent eigenvectors.

6. A matrix is diagonalizable if for all eigenvalues their AM = GM.