Essential Vocabulary

Here is a link to a list of these terms on Quizlet

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Column matrix (or column vector)

A matrix with \(m\) rows and only 1 column.

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Diagonal matrix

A matrix \(A=[a_{ij}]\) where \(a_{ij}=0\) whenever \(i \ne j\)

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Dimensions of a matrix

An \(m \times n\) matrix is a matrix with \(m\) rows and \(n\) columns.

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Elementary matrix

A matrix formed by applying one elementary row operation to the identity matrix.

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Homogeneous system

A system of linear equations is called homogeneous if the system can be written in the form
\[\begin{array}{ccccccccc} a_{11}x_1 &+ &a_{12}x_2&+&\ldots& +&a_{1n}x_n&= &0 \\ a_{21}x_1 &+ &a_{22}x_2&+&\ldots& +&a_{2n}x_n&= &0 \\ &&&&\vdots&&&& \\ a_{m1}x_1 &+ &a_{m2}x_2&+&\ldots& +&a_{mn}x_n&= &0 \end{array},\]
or as a matrix equation as \(A \vec {x} = \vec {0}\).

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Identity matrix

A square matrix with ones as diagonal entries and zeros for the remaining entries.

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Inverse of a square matrix

Let \(A\) be an \(n\times n\) matrix. An \(n\times n\) matrix \(B\) is called an inverse of \(A\) if
\[AB=BA=I\]
where \(I\) is an \(n\times n\) identity matrix. If such an inverse matrix exists, we say that \(A\) is invertible. If an inverse does not exist, we say that \(A\) is not invertible. The inverse of \(A\) is denoted by \(A^{-1}\).

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LU factorization

A factorization \(A=LU\) where \(L\) is lower triangular and \(U\) is upper triangular with ones on the diagonal (called unit upper triangluar). It is useful for solving \(Ax=b\).

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Matrix

A rectangular array of numbers. It has \(m\) rows and \(n\) columns for some positive integers \(m\) and \(n\).

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Matrix addition

Let \(A=\begin{bmatrix} a_{ij}\end{bmatrix} \) and \(B=\begin{bmatrix} b_{ij}\end{bmatrix}\) be two \(m\times n\) matrices. Then the sum of matrices \(A\) and \(B\), denoted by \(A+B\), is an \(m \times n\) matrix given by
\[A+B=\begin{bmatrix}a_{ij}+b_{ij}\end{bmatrix}\]

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Matrix factorization

Representing a matrix as a product of two or more matrices.

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Matrix multiplication

Let \(A\) be an \(m\times n\) matrix whose rows are vectors \(\vec {r}_1\), \(\vec {r}_2,\ldots ,\vec {r}_n\). Let \(B\) be an \(n\times p\) matrix with columns \(\vec {b}_1, \vec {b}_2, \ldots , \vec {b}_p\). Then the matrix product \(AB\) is an \(m \times p\) matrix with entries given by the dot products
\[AB=\begin{bmatrix}-&\vec {r}_1&-\\-&\vec {r}_2&-\\ &\vdots & \\-&\vec {r}_i &-\\ &\vdots& \\-&\vec {r}_m&-\end{bmatrix}\begin{bmatrix}|&|&&|&&|\\\vec {b}_1& \vec {b}_2 &\ldots & \vec {b}_j&\ldots& \vec {b}_p\\|&|&&|&&|\end{bmatrix}=\]
\[=\begin{bmatrix}\vec {r}_1\dotp \vec {b}_1&\vec {r}_1\dotp \vec {b}_2&\ldots& \vec {r}_1\dotp \vec {b}_j&\ldots &\vec {r}_1\dotp \vec {b}_p\\\vec {r}_2\dotp \vec {b}_1&\vec {r}_2\dotp \vec {b}_2&\ldots& \vec {r}_2\dotp \vec {b}_j&\ldots &\vec {r}_2\dotp \vec {b}_p\\\vdots& \vdots&& \vdots&& \vdots \\\vec {r}_i\dotp \vec {b}_1&\vec {r}_i\dotp \vec {b}_2&\ldots& \vec {r}_i\dotp \vec {b}_j&\ldots &\vec {r}_i\dotp \vec {b}_p\\\vdots& \vdots&& \vdots&& \vdots \\\vec {r}_m\dotp \vec {b}_1&\vec {r}_m\dotp \vec {b}_2&\ldots& \vec {r}_m\dotp \vec {b}_j&\ldots &\vec {r}_m\dotp \vec {b}_p\end{bmatrix} \]

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Matrix powers

If \(A\) is a square matrix then we can define \(A^k\) to be the result of multiplying \(A\) by itself \(k\) times.

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Negative of a matrix

The additive inverse of a matrix, formed by multiplying the matrix by the scalar \(-1\).

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Nonsingular matrix

A square matrix \(A\) is said to be nonsingular provided that \(\mbox {rref}(A)=I\). Otherwise we say that \(A\) is singular.

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Partitioned matrices - block multiplication

Subdividing a matrix into submatrices using imaginary horizontal and vertical lines - used to multiply matrices more efficiently.

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Permutation matrix

A matrix formed by permuting the rows of the identity matrix.

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Properties of matrix algebra

Addition and scalar multiplication of matrices have many nice properties. Matrix multiplication is not commutative, but the associative and distributive laws hold. The inverse and transpose operations of a product follow the "shoes and socks" rule.

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Row matrix

A matrix with only \(1\) row and \(n\) columns.

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Scalar multiple of a matrix

If \(A=\begin{bmatrix} a_{ij}\end{bmatrix} \) and \(k\) is a scalar, then \(kA=\begin{bmatrix} ka_{ij}\end{bmatrix}\).

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Singular matrix

A square matrix \(A\) is said to be singular provided that \(\mbox {rref}(A)\) is NOT the identity matrix. If instead \(\mbox {rref}(A)=I\), we say that \(A\) is nonsingular.

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Square matrix

A matrix with the same number of rows and columns.

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Symmetric matrix

An \(n\times n\) matrix \(A\) is said to be symmetric if \(A=A^{T}.\) It is said to be skew symmetric if \(A=-A^{T}.\)

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Transpose of a matrix

Let \(A=\begin{bmatrix} a _{ij}\end{bmatrix}\) be an \(m\times n\) matrix. Then the transpose of \(A\), denoted by \(A^{T}\), is the \(n\times m\) matrix given by switching the rows and columns:
\[ A^{T} = \begin{bmatrix} a _{ij}\end{bmatrix}^{T}= \begin{bmatrix} a_{ji} \end{bmatrix} \]

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Zero matrix

The \(m\times n\) zero matrix is the \(m\times n\) matrix having every entry equal to zero. The zero matrix is denoted by \(O\).

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