What is Orthogonal Diagonalization

  • Remember that in 7.1 Orthogonal and orthonormal bases an orthogonal matrix is an nxn matrix whose columns form an orthonormal basis for

    • From 6.2 Diagonalization
    • A is similar to B if there exists a matrix P such that
    • A is diagonalizable if it is similar to a diagonal matrix D.
  • Suppose we want P to be an orthogonal matrix, not just any invertible matrix this give rise to the following definition
  • Definition

    • We say that two nxn matrices A and B are orthogonally similar if there exists an orthogonal matrix P such that
      • we say that A is orthogonally diagonalizable if A is similar to diagonal matrix D. that is there exists an orthogonal matrix P and diagonal matrix D such that:

When is a matrix orthogonally diagonalizable?

  • Let be matrix ok! to know if a matirx is orthoganlly diagonalizable, we must find if the dot products of the eigenspace is 0

How to orthogonally diagonalize

  • Its less about how to do it, but what the relation ship looks like to the vector

    • You might be asked to find or in the terms of