Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. This theorem is named after two mathematicians, Arthur Cayley & William Rowan Hamilton. This theorem provides an alternative way to find the inverse of a matrix. Let aoλn + a1λn-1 + a2λn-2 + ………………. + an-2λ2 + an-1λ + an = 0 be … Read more
The determinant of matrix P is denoted as |P| i.e. matrix name between two parallel lines. It is also written as det(P) or by symbol delta (Δ). The determinant is always calculated for a square matrix. So if we talk about matrix in this article then it will be understood as a square matrix. In … Read more
A rectangular arrangement of numbers into rows and columns is known as a matrix. The horizontal lines are called rows and the vertical lines are called columns. The matrix is enclosed by [] or (). Let A be any matrix. It can be written as A = [aij]m × n where, aij represents … Read more
The sum of the elements of the principal or main diagonal elements of a square matrix is known as the trace of a matrix. It is generally denoted by Tr(P), where P is any square matrix. Trace of a matrix example Let C is a 2 x 2 matrix such that Now, to … Read more
A matrix obtained as a resultant by changing rows into columns and columns into rows of any matrix is known as the transpose of a matrix. It is generally denoted by PT or P’, where P is any matrix. Transpose of a matrix example Let R is a matrix such that Now, to … Read more
Conjugate of a matrix is the matrix obtained from matrix ‘P’ on replacing its elements with the corresponding conjugate complex numbers. It is denoted by . Conjugate of a matrix example Let Q is a matrix such that Now, to find the conjugate of this matrix Q, we find the conjugate of each … Read more
Conjugate transpose of a matrix ‘P’ is basically a matrix which is equal to the conjugate of the matrix obtained by taking the transpose of the matrix ‘P’. In order to find the conjugate transpose of any matrix; firstly, transpose is obtained and secondly, the conjugate is obtained. The conjugate transpose is generally denoted as … Read more
When the conjugate transpose of a complex square matrix is equal to the inverse of itself, then such matrix is called as unitary matrix. If Q is a complex square matrix and if it satisfies Qθ = Q-1 then such matrix is termed as unitary. Please note that Qθ and Q-1 represent the conjugate transpose and inverse of the matrix Q, … Read more
When the conjugate transpose of a complex square matrix is equal to the negative of itself, then this matrix is called as skew hermitian matrix. If P is a complex square matrix and if it satisfies Pθ = -P then such matrix is termed as skew hermitian. It is noted that Pθ represents the conjugate transpose of matrix P. … Read more
When the conjugate transpose of a complex square matrix is equal to itself, then such matrix is known as hermitian matrix. If B is a complex square matrix and if it satisfies Bθ = B then such matrix is termed as hermitian. Here Bθ represents the conjugate transpose of matrix B. The conjugate transpose of a matrix … Read more
