
The Adjoint of a Matrix


To find the adjoint of a matrix, M, the following transformation is applied: take the transpose of the matrix and then take the complex conjugate of all elements of the matrix. The resulting matrix is called the adjoint of M and is denoted by ${M}^{\mathit{\*}}$.
Note that if all entries of M are real numbers then ${M}^{t}\mathit{\=}\mathit{}{M}^{\mathit{ast;}}$ because each entry is the complex conjugate of itself.
Enter a matrix M =
$\stackrel{\mathrm{take}\mathrm{transpose}}{\to}$$\stackrel{\mathrm{take}\mathrm{conjugate}}{\to}$ = ${\mathrm{M}}^{\*}$




A matrix U is said to be orthogonal if all of its entries are real numbers and ${M}^{1}equals;{M}^{ast;}$, where ${M}^{\*}$denotes the adjoint of M. If the entries of the matrix are complex numbers, M is said to be unitary. An interesting fact is that if a matrix is orthogonal or unitary then its eigenvalues are real numbers and are either 1 or 1.
A matrix N is said to be normal if $N\cdot {N}^{ast;}equals;{N}^{ast;}\cdot N$.
A $n$x$n$ matrix M is said to be orthogonally/unitarily diagonalizable if there exists an orthogonal or unitary $n$x$n$ matrix U such that for a diagonal $n$x$n$ matrix D:
$Mequals;{\mathrm{UDU}}^{1}equals;{\mathrm{UDU}}^{ast;}\phantom{\rule[0.0ex]{0.0em}{0.0ex}}$
This is equivalent to saying: M is similar to a diagonal matrix by using a orthogonal or unitary matrix as a transition matrix.
Is there an easier way to check if a matrix is orthogonally/unitarily diagonalizable?
A matrix A is normal if and only if A is orthogonally/unitarily diagonalizable.
So to check if we can diagonalize the matrix, we must check first if it's normal. This is quite simple from the definition of a normal matrix because it only requires for us to calculate the matrix's adjoint and multiply to verify the condition.
There is, in fact, a procedure in which we can find the diagonal and transition matrices if we determine that the matrix is normal. The procedure is explained step by step below.