When we study eigenvectors of symmetric matrices, there are two important ideas to understand: orthogonality and normalization.
Orthogonality: If a symmetric matrix has different eigenvalues, the eigenvectors that go with them are orthogonal. This means that if you multiply two different eigenvectors together in a special way called the dot product, you will get zero. This property makes calculations easier and helps us work better in spaces with many dimensions.
Normalization: Normalizing eigenvectors just means changing their size so that they each have a length of one. This is useful in areas like machine learning and computer graphics, where having vectors of the same length is very important.
When you put these two ideas together, you create something called an orthonormal basis of eigenvectors. This is really helpful when we want to simplify or diagonalize matrices!
When we study eigenvectors of symmetric matrices, there are two important ideas to understand: orthogonality and normalization.
Orthogonality: If a symmetric matrix has different eigenvalues, the eigenvectors that go with them are orthogonal. This means that if you multiply two different eigenvectors together in a special way called the dot product, you will get zero. This property makes calculations easier and helps us work better in spaces with many dimensions.
Normalization: Normalizing eigenvectors just means changing their size so that they each have a length of one. This is useful in areas like machine learning and computer graphics, where having vectors of the same length is very important.
When you put these two ideas together, you create something called an orthonormal basis of eigenvectors. This is really helpful when we want to simplify or diagonalize matrices!