MatLab: Scratch Pad & Arithmetic Operations

Major Reference Source: MatLab Verson 7.0

Using MatLab package as a scratch pad interactively is one of the common application of the MatLab package. MatLab commands are entered at the prompt command, >>, of the Command Window through the PC keyboard. Both entered commands and evaluation results are displayed in the Command Window. All new line or lines of commands or instructions after the prompt command will be passed to MatLab for evaluation after pressing the Enter key no matter the position of mouse cursor is at the end of line or lines or not.

Arithmetic Operators

Since all types of data in MatLab are stored in the form of arrays, MatLab provides two different kinds of arithmetic operations, i.e. array operations and matrix operations. MatLab array operations are just ordinary arithmetic operations that supports multidimenstional arrays for processing element by elemt operations. While MatLab matrix operation are ordinary matrix operations that following the rules of linear algebra. Both array operations and matrix operations share the similar symbols of operation, a period characater, "." is used to distinguish the array operations from the matrix operations. However, as the addition and subtraction for both matrix operation and array operation are the same, the period characater, "." is not necessary and the character pairs ".+" and ".-" are not used. Besides, the 1-by-1 array, scalar, is also a special type of MatLab array. A 1-by-1 array, scalar can have array operation with an array of any size also. A 1-by-1 matrix, scalar can also have martrix operation with a matrix of any size, but limited by the matrix multiplication, the 1-by-1 matrix, scalar can only be the divisor of the right and left division.

The matrix and array arithmetic includes seven types of operations:

Matrix Arithmetic Operations

+ Addition

The + addition operator of the expression A+B means adds matrix B to matrix A. Since only a scalar can be added by or added to a matrix of any size, unless either A or B is a scalar, A and B must have the same size.

Examples

• m-by-n matrix A + m-by-n matrix B

  

• scalar S + matrix A + scalar S

  

+ Unary Plus

The + unary plus operator of the expression +A means returns matrix A.

Examples

• +matrix A

  

- Subtraction

The - subtraction operator of the expression A-B means subtracts matrix B from matrix A. Since only a scalar can be subtracted by or subtracted from a matrix of any size, unless either A or B is a scalar, A and B must have the same size.

Examples

• m-by-n matrix A - m-by-n matrix B

  

• scalar S - m-by-n matrix A - scalar S

  

- Unary Minus

The - unary minus operator of the expression -A means returns and negates the elements of matrix A.

Examples

• -matrix A

  

* Matrix Multiplication

The * matrix multiplication operator of the expression A*B means to carry out the linear algebraic product of matrix A by matrix B.

Since only a scalar can be multiplicated by or multiplicated to a matrix of any size,  two or more dimensions are supported, unless either A or B is a scalar, otherwise the number of columns of matrix A must equal to the number of rows of matrix B.

Examples

• m-by-n matrix A * n-by-o matrix B

  

• scalar S * m-by-n matrix A * scalar S

  

/ Slash or Matrix Right Division

The / slash or matrix right division operator of the expression A/B means to carry out the linear algebraic product of matrix A by the inverse of matrix B, A*inv(B), in a general sense of the solution to the equation xB=A or A/B equals to (B'\A')'

Limited by the linear algebraic product of matrix, only a scalar can divide a matrix of any size, unless B is a scalar, otherwise both matrices A and B must have the same number of columns.

Examples

• m-by-n matrix A / o-by-n matrix B

  

• matrix A / scalar S

  

\ Backslash or Matrix Left Division

The \ backslash or matrix left division operator of the expression A\B means to carry out the linear algebraic product of the inverse of matrix A by matrix B, inv(A)*B, in a general sense of the solution to the equation Ax=B.

Limited by the linear algebraic product of matrix, only a scalar can divide a matrix of any size, unless A is a scalar, otherwise both matrices A and B must have the same number of rows.

Examples

• m-by-n matrix A \ m-by-o matrix B

  

• scalar S \ matrix A

  

^ Matrix Power

The ^ matrix power operator of the expression A^B means to carry out the matrix A to the power of matrix B. However, one of the matrices A and B must be a scalar and the other matrix must be a square matrix.

If B is a scalar of a positive integer and A is a matrix, the matrix power means repeated matrix multiplication of matrix A by itself of the value of scalar B times. If B is scalar of a negative integer, the matrix power means repeated matrix multiplication of the inverse of matrix A by itself to the absolute value of scalar B times. If B is a scalar not equal to an integer and A is a matrix, the matrix power means the calculation involves eigenvalues and eigenvectors, such that if [V,D]=eig(A), then A^B=V*D.^B/V.

If A is a scalar and B is a matrix, the matrix power also means the calculation involves eigenvalues and eigenvectors, such that if [V,D]=eig(A), then A^B=V*D.^B/V.

Examples

• square matrix A ^ scalar S of positive integer

  

• square matrix A ^ scalar S of negative integer

  

• square matrix A ^ scalar S of non-integer

  

• scalar S ^ square matrix A

  

' Matrix Transpose or Complex Conjugate Transpose

The ' matrix transpose or complex conjugate transpose operator of the expression A' means to carry out the linear algebraic transpose of matrix A. If matrix A is a complex matrix, the matrix transpose operator means to carry out the complex conjugate transpose.

Examples

• m-by-n matrix A'

  

• m-by-n complex matrix A'

  

Array Arithmetic Operations

+ Addition

The + addition operator of the expression A+B means adds B to A. Since only a scalar can be added to a matrix of any size, unless either A or B is a scalar, A and B must have the same size.

Examples

• m-by-n-by-o array A + m-by-n-by-o array B

  

• scalar S + m-by-n array A + scalar S

  

+ Unary Plus

The + unary plus operator of the expression +A means returns A.

Examples

• +array A

  

- Subtraction

The - subtraction operator of the expression A-B means subtracts B from A. Since only a scalar can be subtracted by or subtracted from a matrix of any size, unless either A or B is a scalar, A and B must have the same size.

Examples

• m-by-n-by-o array A - m-by-n-by-o array B

  

• scalar S - m-by-n-by-o array A - scalar S

  

- Unary Minus

The - unary minus operator of the expression -A means returns and negates the elements of A.

Examples

• -array A

  

.* Array Multiplication

The .* array multiplication operator of the expression A.*B means to carry out the element by element product of array A by array B.

Since only a scalar can be multiplicated by or multiplicated to a matrix of any size, unless either A or B is a scalar, otherwise the size of arrays A and B must be equal. Arrays with two or more dimensions are supported.

Examples

• m-by-n-by-o array A .* m-by-n-by-o array B

  

• scalar S .* m-by-n-by-o array A .* scalar S

  

./ Array Right Division

The ./ array right division operator of the expression A./B means to carry out the element by element division of array A by array B.

Since only a scalar can be divided by or divide an array of any size, unless either A or B is a scalar, otherwise the size of arrays A and B must be equal. Arrays with two or more dimensions are supported.

Examples

• m-by-n-by-o array A ./ m-by-n-by-o array B

  

• scalar S ./ m-by-n-by-o array A ./ scalar S

  

.\ Array Left Division

The .\ array left division operator of the expression A.\B means to carry out the element by element division of array B by array A.

Since only a scalar can be divided by or divide an array of any size, unless either A or B is a scalar, otherwise the size of arrays A and B must be equal. Arrays with two or more dimensions are supported.

Examples

• m-by-n-by-o array A .\ m-by-n-by-o array B

  

• scalar S .\ m-by-n-by-o array A .\ scalar S

  

.^ Array Power

The .^ array power operator of the expression A.^B means to carry out the element by element calculation of the element of array A to the corresponding element of array B.

Since only a scalar can be carried out the element by element operation to an array of any size, unless either A or B is a scalar, otherwise the size of arrays A and B must be equal. Arrays with two or more dimensions are supported.

Examples

• m-by-n-by-o array A .^ m-by-n-by-o array B

  

• scalar S .^ m-by-n-by-o array A .^ scalar S

  

.' Array Transpose

The .' array transpose of the expression A.' means to carry out the two dimensional array transpose of array A for both real and complex array and no complex conjugation is involved. Only one or two dimensional arrays are supported

Examples

• m-by-n array A.'

  

• m-by-n complex array A.'

  

Arithmetic Operation Functions

The function of an arithmetic operator also have its corresponding function call.

Arithmetic Expressions

An arithmetic expression is an expression used to represents a numeric value. Arithmetic expressions are built up from numbers, arithmetic operators and parenthese,  etc. following some predefined rules and operator precedence.

Operator Precedence

An expression is usually formed from a combination of arithmetic, relational, and logical operators. Arithmetic operators are the most common operators used for performing arithmetic calculation on numbers. MatLab always evaluates expressions from left to right, however the order of operators to be evaluated by MatLab within an expression are determined by the precedence levels of operators. The following operator with a higher precedence level will override the rule of from left to right evaluation. While the operator will be evaluated immediately if the following operator has the same or lower precedence level. The precedence rules for MatLab operators are

Since parentheses have the highest precedence level, parentheses are used to override the default precedence of all other operations.