Matrix Calculator

Welcome to Omni's matrix calculator! This humongous matrix solver serves as a hub to connect and coordinate all of Omni's calculators that involve various matrix operations in math. Here, you can get a bird's eye view of the wide matrix landscape:

• Learn (or recall) what a matrix is in math;
• What the most important matrix types are, and
• Find a large collection of links to (almost) all of our matrix calculators.

Enjoy!

A matrix is a fancy name for an array of numbers. An example of a matrix would be

Matrices have rows and columns. In matrix $A$ above, the 1st row is $[1\ 2]$ and the 2nd row is $[3\ 4]$. Its 1st column and 2nd columns read respectively

The number of rows and columns gives the dimensions of the matrix. In our example, $A$ is a $2 \times 2$ matrix with two rows by two columns.

Based on this dimension, we distinguish several types of matrices:

• For a square matrix, the number of rows equals the number of columns.
• A row matrix has only one row (hence the name) and several columns.
• A column matrix has only one column and several rows.

Moreover, we say that a matrix has cells in which we write the elements of our array. For example, the cell in the 2nd row and the 1st column of $A$ contains the value $3$: the coordinates of this cell is $(2,1)$ and we would notate it as $a_{2,1} = 3$.

Matrices are a convenient way to store and manipulate more data than just an individual number. And so now the question arises: what operations can we perform on matrices? Can we for instance add or multiply them like ordinary numbers?

Well, almost. Since lots of numbers are involved simultaneously, calculations with matrices are more tricky than with individual numbers. For instance, if we want to add them, we first have to make sure that we can — only matrices with the same dimensions can be added. And for multiplication, the dimension requirement is even trickier.

This matrix calculator is very straightforward to use. Here's how:

1. The first few fields at the top of our matrix calculator help you pick the matrix operation you need. They are sorted logically, but don't worry — the whole list is next in the following section.
2. Choose the matrix size. In this matrix solver only the dimensions $2\times2$ and $3\times3$ are available. More sizes are available in the calculators dedicated to the selected matrix operation — the specific link is displayed at the bottom.
3. Enter the coefficients of your matrix and enjoy the result that appears immediately.
4. For more info on the matrix operation you've just performed, visit the dedicated tool.

Here we list all the matrix math operations available in our matrix solver. To discover more about them, follow the links to the dedicated calculators.

Math operations that act on one matrix (unary matrix operations)

• Operations that return a number:

  • Trace;
  • Determinant;
  • Rank; and
  • Matrix norms.

• Operations that return a matrix:

  • Inverse matrix;
  • Pseudoinverse (Moore Penrose inverse) matrix;
  • Adjoint (adjugate) matrix;
  • Cofactor matrix;
  • Power of a matrix;
  • Transpose;
  • Multiplication by a number;
  • Diagonalization.

• Decompositions:

  • LU decomposition;
  • Singular values decomposition (SVD);
  • QR decomposition;
  • Cholesky decomposition; and
  • Polar decomposition.

• Other:

  • Eigenvalues and eigenvectors;
  • Characteristic polynomial;
  • Singular values; and
  • Detecting matrix type (see the next section).

Math operations that act on two matrices (binary matrix operations)

• Addition and subtraction;
• Multiplication (matrix by matrix);
• Kronecker (tensor) product; and
• Hadamard (entrywise) product.

The most popular special types of matrices are the following:

• Diagonal;
• Identity;
• Triangular (upper or lower);
• Symmetric;
• Skew-symmetric;
• Invertible;
• Orthogonal;
• Positive/negative definite; and
• Positive/negative semi-definite.

Let's briefly define each of the matrix types that we mentioned above.

• Diagonal matrix

  Square matrices that have non-zero coefficients only in the diagonal cells. It's very easy to compute its powers.

• Identity matrix

  This is a diagonal matrix that has only ones on its diagonal and zeroes everywhere else. It's everyone's favorite matrix when it comes to matrix multiplication, because it leaves the other matrix unchanged — much like multiplying a number with $1$!

• Triangular matrix (upper or lower)

  A square matrix with non-zero coefficients on the diagonal and above the diagonal (if it's upper triangular) or below the diagonal (if lower triangular).

  Its determinant coincides with the product of the diagonal values. Often appears in matrix decompositions and numerical methods.

• Symmetric matrix

  A square matrix that is symmetric with respect to its diagonal, i.e., $a_{j,i}=a_{i,j}$ for all $i,j=1, \ldots, n$. In words: the coefficient in $i$-th row and $j$-th column equals the coefficient of $j$-th row and $i$-th column. Such a matrix has real eigenvalues and an orthonormal eigenbasis.

• Skew-symmetric (antisymmetric) matrix

  A square matrix whose entries satisfy $a_{ji}=-a_{ij}$. It follows that the diagonal elements are all equal to zero, because only $a_{i,i}=0$ can satisfy $a_{i,i} = -a_{i,i}$. The trace of an antisymmetric matrix is therefore always equals zero.

• Invertible matrix

  A square matrix that has an inverse, i.e., $A$ is invertible if there exists $B$ such that $AB = BA = I$, where $I$ is the identity matrix. The determinant of an invertible matrix is always non-zero.

• Orthogonal matrix

  This is a square matrix whose columns constitute a set of orthonormal vectors (and its vectors form such a set as well). Equivalently we can say that a matrix is orthogonal if its transpose coincides with its inverse. The determinant of an orthogonal matrix is equal to $1$ or $-1$.

• Definite matrices

  All the matrices we consider below are symmetric (or Hermitian). $x^{T}$ denotes the (Hermitian) transpose of $x$ (whether $x$ is a vector or a matrix).

  • Positive semi-definite matrix
    A matrix $A$ is positive semi-definite if $x^{{T}}Ax\geq 0$ for every vector $x$.

    Only positive semi-definite matrices have real and non-negative eigenvalues, and all positive semi-definite matrices have such eigenvalues.

  • Positive definite matrix
    A matrix $A$ is positive definite if $x^{ {T}}Ax > 0$ for every non-zero vector $x$.

    Only positive definite matrices have real and positive eigenvalues, and all positive definite matrices have such eigenvalues.

  • Negative semi-definite matrix
    A matrix $A$ is negative semi-definite if $x^{ {T}}Ax\leq 0$ for every vector $x$.

    Negative semi-definite matrices are exactly those matrices whose eigenvalues are all real and non-positive.

  • Negative-definite matrix
    A matrix $A$ is negative definite if $x^{ {T}}Ax < 0$ for every non-zero vector $x$.

    Negative definite matrices are exactly those matrices whose eigenvalues are all real and negative.

Certain types of matrices in maths are easy to detect, while other types are more intricate. Here are a few tips on how to detect matrix type:

1. Start by just looking at the matrix structure: Is it diagonal? Symmetric? Triangular?
2. Other types depend on deeper properties — check the eigenvalues, inverse, transpose, product of transpose and the initial matrix, etc.
3. Use advanced math software, e.g., Omni's matrix solver, to help you detect the type of your matrix.

No one really knows, but there's a lot of them. Scientists come up with new matrix operations which help them deal with different problems related to matrices and their applications in real life. Remember that matrix operations can act on one or more matrices, and they can return another matrix, multiple matrices, or a single number/vector, or a set of numbers/vectors or a polynomial, etc. The possibilities are endless!

Because matrices are a perfect way to deal with lots of numbers simultaneously, they are extremely useful in many domains of our modern, data-laden life. Real-life matrix applications include:

• 3D graphics, e.g. in games;
• Cryptography and data science;
• Economics and econometrics;
• Engineering and construction;
• Electronics, and
• Physics in general.