A sequence is a fundamental arrangement of things, objects, numbers, etc. The sequences may be finite or infinite depending on the number of terms. There are many types of sequences such as geometric sequence, harmonic sequence, and arithmetic sequence which are used to perform the requisite operation. In this topic, we only discuss the arithmetic … Read more
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 \[A=[a_{ij}]_{m\times n}=\begin{bmatrix} a_{11} & a_{12} … 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 \[C=\begin{bmatrix} 3 &4 \\ … 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 \[R=\begin{bmatrix} 1 & 2 … 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 $\overline{P}$. Conjugate of a matrix example Let Q is a matrix such that \[Q=\begin{bmatrix} 1+i &2+3i \\ 4-2i& 6 \end{bmatrix}\] Now, to find the conjugate of this matrix Q, we … 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