Is Orthogonality Preserved If Inner Product Changes?

In this video, we talk about how defining a new inner product for a vector space can influence the orthogonality of two vectors that are orthogonal under the dot product. The SPD matrix that defines an inner product in an inner product space can significantly change the geometry of the space by stretching or decompressing the space in a specific direction. We also introduce the concepts of orthogonality and orthonormality and define what an orthogonal matrix is. Basically, an SPD matrix gives us a new inner product and orthogonal matrices preserve inner products.