Operations on matrices are the fundamental ways of combining and transforming matrices. The main operations include:
- Addition: $A + B$ — entrywise sum (requires same size)
- Scalar multiplication: $cA$ — multiply each entry by scalar $c$
- Matrix multiplication: $AB$ — row-by-column dot products (requires compatible dimensions)
- Transpose: $A^T$ — flip rows and columns
- Conjugate transpose (adjoint): $A^*$ or $A^H$ — transpose + complex conjugate
- Inverse: $A^{-1}$ — matrix that satisfies $AA^{-1} = I$ (for nonsingular matrices)
- Row operations: elementary operations used in Gaussian elimination
These operations make the set of $m \times n$ matrices into a vector space, and the set of $n \times n$ matrices into a ring.
Parents
Children
- AdditionDefinition: The sum of two matrices $A$ and $B$ of the same size ($m \times n$) is the matrix $A + B$ defined by:...
- ConjugationDefinition: The complex conjugate of a matrix $A = (a {ij})$ with complex entries is the matrix $\overline{A}$...
- Scalar multiplicationDefinition: The scalar multiplication of a matrix $A = (a {ij})$ by a scalar $c$ is the matrix $cA$ defined by:...
- Row operationsDefinition: Row operations (elementary row operations) are three fundamental operations that can be performed on the...
- Matrix-vector productsDefinition: The matrix-vector product $Ax$ is the product of an $m \times n$ matrix $A$ and a column vector $x \in...
- MultiplicationDefinition: The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $C =...
- Transpose and adjointTranspose: The transpose of an $m \times n$ matrix $A$, denoted $A^T$, is the $n \times m$ matrix obtained by...
- InverseDefinition: A square matrix $A$ is invertible (or nonsingular ) if there exists a matrix $A^{-1}$ such that:...
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