From my understanding, the inverse matrix is equivalent to its transpose because (1) the columns are unit length and (2) the columns are at a right angle of each other (i.e. dot product = 0). This means we have an orthonormal matrix and an orthonormal matrix's inverse is equivalent to its transpose.

simonguozirui

Professor mentioned the inverse is just the transpose if columns are orthogonal to each other. Does it also have to be normalized because this property seem to be only true for orthonormal basis. Also is it possible to have an axis coordinate frame / basis that is not orthogonal (like spherical cooridnates)?

edit: oops I see professor briefly mention about non-orthogonal basis later.

Staffjamesfobrien

Orthonormal means orthogonal and each is unit length (normalized). Consider A A^T where A is orthonormal. You should be able to show the identity matrix.

From my understanding, the inverse matrix is equivalent to its transpose because (1) the columns are unit length and (2) the columns are at a right angle of each other (i.e. dot product = 0). This means we have an orthonormal matrix and an orthonormal matrix's inverse is equivalent to its transpose.

Professor mentioned the inverse is just the transpose if columns are orthogonal to each other. Does it also have to be normalized because this property seem to be only true for orthonormal basis. Also is it possible to have an axis coordinate frame / basis that is not orthogonal (like spherical cooridnates)?

edit: oops I see professor briefly mention about non-orthogonal basis later.

Orthonormal means orthogonal and each is unit length (normalized). Consider A A^T where A is orthonormal. You should be able to show the identity matrix.