Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

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What does it mean that the eigenvalues of a matrix are the zeros of its characteristic polynomial? And that the matrix itself is a zero of its characteristic polynomial? And then what doe the symmetric functions of the eigenvalues of a matrix mean? The coefficients of the polynomial can be expressed in terms of the same eigenvalues that are its solutions. So in what sense are they dual? Ask at Math Overflow.

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In solving for eigenvalues {$\lambda_i$} and eigenvectors {$v_i$} of {$M$}, make the matrix {$M-\lambda I$} degenerate. Thus {$\text{det}(M-\lambda I)=0$}. The matrix is degenerate when one row is a linear combination of the other rows. So the determinant is a geometrical expression for volume, for collinearity and noncollinearity.
An inner product on a vector space allows it to be broken up into vector spaces that complement each other, thus into irreducible vector spaces.
Matrices {$A=PBP^{-1}$} and {$B=P^{-1}AP$} have the same eigenvalues. They are simply written in terms of different coordinate systems. If v is an eigenvector of A with eigenvalue λ, then {$P^{-1}v$} is an eigenvector of B with the same eigenvalue λ.
Eigenvectors are the pure dimensions into which the action of a matrix (or linear transformation) can be decomposed.

Prove that the matrix made up of eigenvectors diagonalizes a matrix.

Ravi Vakil: Main theme of mathematics - convert harder problems to linear algebra