Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 7B (Vectors and Matrices)

The matrix trace isUsing matrix multiplication and interchanging dummy indices,

The proposed noncyclic interchange is false. With matrix units,one hasTrace is invariant under cyclic permutations, but an arbitrary swap need not be cyclic.

The characteristic polynomial of $A$ isA scalar $\lambda \in \mathbb{C}$ is an eigenvalue when there is a nonzero vector $v$ such thatSuch a $v$ is an eigenvector belonging to $\lambda$. Equivalently, $\lambda$ is a root of ${\chi }_A$ and the corresponding eigenvectors are the nonzero elements of $\ker (A-\lambda I)$.

Consider the real polynomialThe hypothesis says $p(i)\ne 0$, so $p$ is not identically zero. A nonzero polynomial has only finitely many roots, whereas $\mathbb{R}$ is infinite. Hence there is a real $\lambda$ withThus $A+\lambda B$ is invertible, as summarized by the real parameter avoiding a singular matrix pencil.

Write the complex similarity matrix aswith $X,Y$ real. From $C=SD{S}^{-1}$ we obtain $CS=SD$, and equating real and imaginary parts givesThereforefor every real $t$.

Now $p(t)=\det (X+tY)$ is a real polynomial andThus some real $t$ has $p(t)\ne 0$. Taking $T=X+tY$, we have a real invertible matrix satisfying $CT=TD$, orThis proves real similarity from complex similarity.

Eigenvectors belonging to distinct eigenvalues are linearly independent. Indeed, supposeis a shortest nontrivial relation, where $A{v}_j={\lambda }_j{v}_j$ and the ${\lambda }_j$ are distinct. Applying $A-{\lambda }_{m}I$ givesa shorter nontrivial relation, a contradiction.

Thus $n$ distinct eigenvalues provide a basis of $n$ eigenvectors, proving that $A$ is diagonalizable over $\mathbb{C}$. If $A$ is real and all its eigenvalues are real, each real singular matrix $A-\lambda I$ has a nonzero real null vector. Choosing one for each eigenvalue gives a real eigenbasis. Hence the real matrix is diagonalizable over $\mathbb{R}$. This is the distinct eigenvalues imply diagonalizability criterion.