BIZNIQUE / LINEAR ALGEBRA / 38
Resolve a symmetric map into independent directions
Resolve a symmetric map into independent directions
In a suitable basis, a complicated-looking matrix can act as separate stretches.
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THE IDEA
Read the example.
Read the example.
A real symmetric matrix has an orthonormal eigenbasis. Q contains its eigenvectors and Λ contains its eigenvalues. Transforming into that basis, applying the separate scales and transforming back reproduces A.
A = QΛQᵀ for real symmetric A
Use it well
Use it well
The orthogonal decomposition stated here is for symmetric matrices. A general matrix may need a different decomposition and may not be diagonalisable.
TRY IT YOURSELF
A quick check.
A quick check.
For A = [[3, 1], [1, 3]], what are the eigenvalues?
Check your reasoning
Check your reasoning
They are 4 along (1, 1) and 2 along (1, −1).
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