Skip to content

An eigen-property of symmetric matrices


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


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.

No comments yet

Leave a Reply

Please log in using one of these methods to post your comment: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: