A useful factorization
A diagonalization writes with D diagonal and P invertible. Powers of D are calculated entry by entry.
Algebra / English
✦ Diagonalization / Diagonalizing a Matrix, Part 1 ✦
Patrick J · YouTube · 7:25
YouTube · Patrick JOriginal video
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Explain diagonalization, simplify powers and state the independent-eigenvector criterion.
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0:00A useful factorization3:00Cancel adjacent factors6:00Enough independent eigenvectorsLearning script
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A diagonalization writes with D diagonal and P invertible. Powers of D are calculated entry by entry.
In repeated products, P inverse followed by P is the identity. Therefore , and the same reasoning works for any positive integer power.
Over the chosen field, an n-by-n matrix is diagonalizable exactly when it has n linearly independent eigenvectors. Put those vectors into P and their corresponding eigenvalues into D in matching order.
Knowledge cards
Cancel adjacent factors
In repeated products, P inverse followed by P is the identity. Therefore , and the same reasoning works for any positive integer power.
Enough independent eigenvectors
Over the chosen field, an n-by-n matrix is diagonalizable exactly when it has n linearly independent eigenvectors. Put those vectors into P and their corresponding eigenvalues into D in matching order.
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- Diagonalization ExplanationAt 0:00
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A diagonalization writes with D diagonal and P invertible. Powers of D are calculated entry by entry. In repeated products, P inverse followed by P is the identity. Therefore , and the same reasoning works for any positive integer power. Over the chosen field, an n-by-n matrix is diagonalizable exactly when it has n linearly independent eigenvectors. Put those vectors into P and their corresponding eigenvalues into D in matching order.
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