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For the given real matrix $M$, direct matrix multiplication gives the eigenvectorsPut these eigenvectors into the columns ofSince $\det P=-2\ne 0$, $P$ is an invertible matrix. The three column identities give $MP=PD$, soIn particular, the matrix similarity is established by an explicit change-of-basis matrix.

The three elementary row operations on an $m\times n$ real matrix are:

Each operation changes whole rows, so it acts on every column by the same rule. Each can be undone by an elementary row operation.

A scalar $\lambda$ is an eigenvalue of $A$ if and only if there is a nonzero vector $v$ with $Av=\lambda v$. By the preceding kernel criterion,The last condition is ${\chi }_A(\lambda )=0$, exactly the definition of a polynomial root. For a real matrix, this proves the assertion for real eigenvalues; the same proof over the complex numbers gives the usual complex version.

A real matrix is a matrix whose entries are real numbers. The vector space of $m\times n$ real matrices is denoted $M_{m,n}(\mathbb{R})$.