Monthly Archives: November 2020

What Is a (Non)normal Matrix?

An matrix is normal if , that is, if commutes with its conjugate transpose. Although the definition is simple to state, its significance is not immediately obvious. The definition says that the inner product of the th and th columns … Continue reading

Posted in what-is | Leave a comment

What Is the Matrix Logarithm?

A logarithm of a square matrix is a matrix such that , where is the matrix exponential. Just as in the scalar case, the matrix logarithm is not unique, since if then for any integer . However, for matrices the … Continue reading

Posted in what-is | Tagged | Leave a comment

What Is a QR Factorization?

A QR factorization of a rectangular matrix with is a factorization with orthonormal and upper trapezoidal. The factor has the form , where is and upper triangular. Partitioning conformably with we have There are therefore two forms of QR factorization: … Continue reading

Posted in what-is | Leave a comment

What is the Cayley–Hamilton Theorem?

The Cayley–Hamilton Theorem says that a square matrix satisfies its characteristic equation, that is where is the characteristic polynomial. This statement is not simply the substitution “”, which is not valid since must remain a scalar inside the term. Rather, … Continue reading

Posted in what-is | Leave a comment