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An eigen-property of symmetric matrices

30/07/2011

Property: For a symmetric matrix A, eigenvectors belong to different eigenspaces are mutually orthogonal.

Proof:

Suppose we have

Ax_1=\lambda_1 x_1
Ax_2=\lambda_2 x_2

and \lambda_1\ne\lambda_2.

Then x_2^T Ax_1=x_2^T \lambda_1 x_1. Note x_2^T Ax_1=(Ax_2)^T x_1=\lambda_2x_2^T x_1. So

\lambda_2 x_2^Tx_1=\lambda_1 x2^T x_1.

Because \lambda_1\ne\lambda_2, so x_2^Tx_1=0 which implies x_2 \perp x_1.

Remark: for complex and self-adjoint matrix, the property still hold.

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