Diagonal scaling
A diagonal matrix scales each coordinate independently. It has an inverse obtained by reciprocal diagonal entries only when every diagonal entry is nonzero.
Algebra / English
Visualizing Diagonalization & Eigenbases
Dr. Trefor Bazett · YouTube · 9:45
YouTube · Dr. Trefor BazettOriginal video
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Visualize diagonalization as a change of coordinates, a scaling, and a return to the original coordinates.
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0:00Diagonal scaling4:30Build an eigenbasis7:30Read the composition right to leftLearning script
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A diagonal matrix scales each coordinate independently. It has an inverse obtained by reciprocal diagonal entries only when every diagonal entry is nonzero.
When enough independent eigenvectors form the columns of P, the matching eigenvalues form the diagonal of D, giving . This condition is essential.
First apply P inverse to obtain eigenbasis coordinates, then D to scale those coordinates, and finally P to return to the original basis. Not every matrix admits this diagonalization.
Knowledge cards
Build an eigenbasis
When enough independent eigenvectors form the columns of P, the matching eigenvalues form the diagonal of D, giving . This condition is essential.
Read the composition right to left
First apply P inverse to obtain eigenbasis coordinates, then D to scale those coordinates, and finally P to return to the original basis. Not every matrix admits this diagonalization.
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- Diagonalization ExplanationAt 0:00
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A diagonal matrix scales each coordinate independently. It has an inverse obtained by reciprocal diagonal entries only when every diagonal entry is nonzero. When enough independent eigenvectors form the columns of P, the matching eigenvalues form the diagonal of D, giving . This condition is essential. First apply P inverse to obtain eigenbasis coordinates, then D to scale those coordinates, and finally P to return to the original basis. Not every matrix admits this diagonalization.
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A different approach
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