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Matrix Calculator Pro Description

A practical math tool to calculate real matrice or complex matrice

How to use it?

Step1: Input Matrix A or B in the left panel,? according to the format showed in the interface;

Step2: Input coefficient x and select precision when necessary, otherwise they equal to the default values.

Step3:Put one of the expression buttons in the middle panel to calculate. The results will be shown in the right panel.

*Support Polar format?
*How to input complex matrix?
*How to calculate complex matrix?

You can input complex number such as 3+i, 3+3i,-3.4i, etc. But not 3+3*i, 3i+3, etc.

For example: Matrix A=

1 -2i 3+i

2 1+i 1-i

i 3.2 14i

Then inv(A)=

0.433-0.182i 0.352+0.077i 0.028+0.136i

-0.215+0.475i 0.062-0.204i -0.067-0.09i

-0.139-0.036i 0.021+0.009i 0.019-0.097i

Polar format is also supported! And the angles are in degrees.

eg.

Matrix A=

1 2(-90) 3.162(18.435)

2 1.414(45) 1.414(-45)

1 3.2 14(90)

Click 'Complex Format' to choose the format of complex or polar.

Notice: In some countries, comma is the default decimal separator. eg:

Matrix A=

1,2 3,2+i

2,1 3+4,2i

Invertible matrix

In linear algebra, an n-by-n (square) matrix A is called invertible or nonsingular or nondegenerate if there exists an n-by-n matrix B such that

AB=BA=I

where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A−1. It follows from the theory of matrices that if

AB=I

for square matrices A and B, then also

BA=I

Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, then A has a left inverse: an n-by-m matrix B such that BA = I. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB = I.

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix, it will almost surely not be singular.

While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any commutative ring. However, in this case the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a much stricter requirement than being nonzero.

Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

LU decomposition

In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear equations or calculate the determinant.

Let A be a square matrix. An LU decomposition is a decomposition of the form

A=LU

where L and U are lower and upper triangular matrices (of the same size), respectively. This means that L has only zeros above the diagonal and U has only zeros below the diagonal.

An LDU decomposition is a decomposition of the form

A=LDU

where D is a diagonal matrix and L and U are unit triangular matrices, meaning that all the entries on the diagonals of L and U are one.

An LUP decomposition (also called a LU decomposition with partial pivoting) is a decomposition of the form

PA=LU

where L and U are again lower and upper triangular matrices and P is a permutation matrix, i.e., a matrix of zeros and ones that has exactly one entry 1 in each row and column.

An LU decomposition with full pivoting (Trefethen and Bau) takes the form

PAQ=LU

Above we required that A be a square matrix, but these decompositions can all be generalized to rectangular matrices as well. In that case, L and P are square matrices which each have the same number of rows as A, while U is exactly the same shape as A. Upper triangular should be interpreted as having only zero entries below the main diagonal, which starts at the upper left corner.

Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication.

The ordinary matrix product is the most often used and the most important way to multiply matrices. It is defined between two matrices only if the width of the first matrix equals the height of the second matrix. Multiplying an m×n matrix with an n×p matrix results in an m×p matrix. If many matrices are multiplied together, and their dimensions are written in a list in order, e.g. m×n, n×p, p×q, q×r, the size of the result is given by the first and the last numbers (m×r), and the values surrounding each comma must match for the result to be defined. The ordinary matrix product is not commutative.

The first coordinate in matrix notation denotes the row and the second the column; this order is used both in indexing and in giving the dimensions. The element Xij at the intersection of row i and column j of the product matrix is the dot product (inner product) of row i of the first matrix and column j of the second matrix. This explains why the width and the height of the matrices being multiplied must match: otherwise the dot product is not defined.

Function:

* Matrix inverse.
* Matrix transpose.
* Largest component.
* Smallest component.
* Matrix or vector norm.
* Dimension.
* Sum of diagonal elements.
* Determinant.
* Matrix rank.
* Eigenvalues.
* QR factorization. Orthogonal-triangular decomposition.
* LU factorization.
* Cholesky factorization.
* Singular value decomposition.
* Plus.
* Minus.
* Matrix multiply.
* Array multiply.
* Slash or right matrix divide.
* Right array divide.
* Element-wise logical AND.
* Element-wise logical OR.
* Logical EXCLUSIVE OR
* Matrix power

Matrix Calculator Pro Version History

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