A symmetric matrix is a square matrix where the elements on either side of the main diagonal are equal. In other words, a symmetric matrix is equal to its own transpose.
To determine if a matrix is symmetric, you can check if it is equal to its transpose. If all the corresponding elements on either side of the main diagonal are equal, then the matrix is symmetric.
An orthogonal projection is a type of linear transformation that projects a vector onto a subspace perpendicular to a given subspace. In other words, it finds the component of a vector that is perpendicular to a given subspace.
To calculate an orthogonal projection, you can use the formula projvw = (w · v / v · v) * v, where w is the vector you want to project, v is the vector defining the subspace, and · represents the dot product.
Some real-world applications of symmetric matrices and orthogonal projections include image and signal processing, data compression, and machine learning algorithms such as principal component analysis (PCA).