Example: Apply the Gram-Schmidt process to find an orthonormal matrix that diagonalizes
Solution
The characteristic polynomial of the matrix is given by
\(= (\lambda + 2)(\lambda^{2} - 2\lambda - {\rm 8})\)
Solving for , we obtain ,
Note that is the eigenvalue of multiplicity
Next, we find the eigenvectors associated with by solving the homogeneous system
Writing the system above in an augmented matrix and applying elementary row operations, we have
which implies
Choosing we get
Thus, basis for the solution space of the linear system consists of the eigenvectors
Since and are not orthogonal, we use the Gram-Schmidt process to obtain an orthonormal basis.
Let
The set is orthogonal.
Normalizing , we obtain
Now, we find the basis for the eigenvector associated with by solving the homogeneous system
Writing the homogeneous system above in augmented matrix and applying elementary row operations
If we choose
Hence a basis for the solution space consists of the eigenvector
Normalizing we have
Therefore, the orthonormal matrix that diagonalizes is
-
- Active Topics
-
-
- by Eli 1 day ago All in One: YouTube, TED, X, Facebook and Instagram Reels, Videos, Images and Text Posts View the latest post Replies 324 Views 29042
- by Eli 1 day ago Re: What is in Your Mind? View the latest post Replies 700 Views 294477
- by Eli 3 days ago Russia Invades Ukraine View the latest post Replies 661 Views 230116
- by Eli 4 days ago President Museveni's Speech During International Development Association (IDA) Summit View the latest post Replies 1 Views 217
- by Eli 4 days ago From Simple Linear Regression Analysis to Covariance & Correlation to Independent Determinant, and R-Squared View the latest post Replies 11 Views 24496
- by Eli 1 week ago Collection of Greatest Christian Hymns of all Times View the latest post Replies 34 Views 62560
- by Eli 1 week ago Pondering Big Cosmology Questions Through Lectures and Dialogues View the latest post Replies 34 Views 57778
- by Eli 1 week ago Programmatically Manipulate Files: Renaming, Reading, Writing, Deleting, and Moving Files Between Folders View the latest post Replies 7 Views 11438
- by Eli 2 weeks ago Iran Launches Retaliatory Attack Against Israel, and Israel Retaliates by Attacking Iranian Isfahan Millitary Base View the latest post Replies 28 Views 11254
- by Eli 2 weeks ago Python Packages for Scientific Computing View the latest post Replies 8 Views 17640
-
Matrix Orthonormal Diagonalization
-
- Information
-
Who is online
Users browsing this forum: No registered users and 0 guests