inv

Inverse of symbolic matrix

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Syntax

D = inv(A)

Description

example

D = inv(A)returns the inverse of a symbolic matrixA。

Examples

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Compute Inverse of Symbolic Matrix

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Compute the inverse of a matrix of symbolic numbers.

A = sym([2 -1 0; -1 2 -1; 0 -1 2]); D = inv(A)

D =
                       
                         $(\begin{array}{c} \frac{3}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{2} & 1 & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{2} & \frac{3}{4} \end{array})$

Compute the inverse of a matrix of symbolic scalar variables.

symsabcdA = [a b; c d]; D = inv(A)

D =
                       
                         $(\begin{array}{c} \frac{d}{a d - b c} & - \frac{b}{a d - b c} \\ - \frac{c}{a d - b c} & \frac{a}{a d - b c} \end{array})$

Compute Inverse of Hilbert Matrix with Symbolic Numbers

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Compute the inverse of the Hilbert matrix that contains symbolic numbers.

D = inv(sym(hilb(4)))

D =
                       
                         $(\begin{array}{c} 16 & - 120 & 240 & - 140 \\ - 120 & 1200 & - 2700 & 1680 \\ 240 & - 2700 & 6480 & - 4200 \\ - 140 & 1680 & - 4200 & 2800 \end{array})$

Compute Inverse of Block Matrix

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Find the inverse of a 4-by-4 block matrix

$C = [\begin{array}{c} A & 0_{2, 2} \\ 0_{2, 2} & B \end{array}]$

where $A$ and $B$ are 2-by-2 submatrices. The notation $0_{2, 2}$ represents a 2-by-2 submatrix of zeros.

Use symbolic matrix variables to represent the submatrices in the block matrix.

symsAB[2 2]matrixZ = symmatrix(zeros(2))

Z =
                        $0_{2, 2}$

C = [A Z; Z B]

C =
                       
                         $(\begin{array}{c} \begin{array}{c} A & 0_{2, 2} \end{array} \\ \begin{array}{c} 0_{2, 2} & B \end{array} \end{array})$

Find the inverse of the matrix $C$ 。

D = inv(C)

D =
                       
                         ${(\begin{array}{c} \begin{array}{c} A & 0_{2, 2} \end{array} \\ \begin{array}{c} 0_{2, 2} & B \end{array} \end{array})}^{- 1}$

To show the elements of the inverse matrix, convert the result from a symbolic matrix variable to symbolic scalar variables usingsymmatrix2sym。

D1 = symmatrix2sym(D)

D1 =
                       
                         $\begin{array}{l} (\begin{array}{c} \frac{A_{2, 2}}{σ_{2}} & - \frac{A_{1, 2}}{σ_{2}} & 0 & 0 \\ - \frac{A_{2, 1}}{σ_{2}} & \frac{A_{1, 1}}{σ_{2}} & 0 & 0 \\ 0 & 0 & \frac{B_{2, 2}}{σ_{1}} & - \frac{B_{1, 2}}{σ_{1}} \\ 0 & 0 & - \frac{B_{2, 1}}{σ_{1}} & \frac{B_{1, 1}}{σ_{1}} \end{array}) \\ where \\ σ_{1} = B_{1, 1} B_{2, 2} - B_{1, 2} B_{2, 1} \\ σ_{2} = A_{1, 1} A_{2, 2} - A_{1, 2} A_{2, 1} \end{array}$

Compute Inverse of Matrix Polynomial

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Compute the inverse of the matrix polynomial $a_{0} I_{2} + a_{1} A + a_{2} A^{2}$ , where $A$ is a 2-by-2 matrix.

Create the matrix $A$ and the coefficients $a_{0}$ , $a_{1}$ , and $a_{2}$ as symbolic matrix variables. Create the matrix polynomial as a symbolic matrix functionfwith $A$ , $a_{0}$ , $a_{1}$ , and $a_{2}$ as its parameters.

symsA[2 2]matrixsymsa0a1a2[1 1]matrixsymsf(A,a0,a1,a2)[2 2]matrixkeepargsf(A,a0,a1,a2) = a0*eye(2) + a1*A + a2*A^2

f(A, a0, a1, a2) =
                        $a_{0} I_{2} + a_{1} A + a_{2} A^{2}$

Find the inverse offusinginv。The result is a symbolic matrix function of typesymfunmatrix。

fInv = inv(f)

fInv(A, a0, a1, a2) =
                        ${(a_{0} I_{2} + a_{1} A + a_{2} A^{2})}^{- 1}$

Evaluate the inverse for the matrix value $A = [2 - 1; - 1 2]$ and the coefficient values $a_{0} = - 1$ , $a_{1} = 2$ , and $a_{2} = 3$ 。The result is a symbolic matrix variable of typesymmatrix。

Aval = [2 -1; -1 2]; fEval = fInv(Aval,-1,2,3)

fEval =
                       
                         $\begin{array}{l} {(2 Σ_{1} + 3 {Σ_{1}}^{2} - I_{2})}^{- 1} \\ where \\ Σ_{1} = (\begin{array}{c} 2 & - 1 \\ - 1 & 2 \end{array}) \end{array}$

Convert the result from thesymmatrixdata type to thesymdata type usingsymmatrix2sym。结果是一个矩阵的年代ymbolic numbers.

symmatrix2sym(fEval)

ans =
                       
                         $(\begin{array}{c} \frac{9}{64} & \frac{7}{64} \\ \frac{7}{64} & \frac{9}{64} \end{array})$

Input Arguments

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`A`—Input matrix
square numeric matrix|square matrix of symbolic scalar variables|square symbolic matrix variable|square symbolic matrix function|symbolic expression with square size

Input matrix, specified as a square numeric matrix, square matrix of symbolic scalar variables, square symbolic matrix variable, square symbolic matrix function, or symbolic expression with square size.

Data Types:single|double|sym|symmatrix|symfunmatrix

Tips

Matrix computations involving many symbolic variables can be slow. To increase the computational speed, reduce the number of symbolic variables by substituting the given values for some variables.

Version History

之前介绍过的R2006a

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R2022a: Compute inverse of symbolic matrix functions

Theinvfunction accepts an input argument of typesymfunmatrix。For an example, seeCompute Inverse of Matrix Polynomial。

R2021b: Compute inverse of symbolic matrix variables

Theinvfunction accepts an input argument of typesymmatrix。For an example, seeCompute Inverse of Block Matrix。

inv

Syntax

Description

Examples

Compute Inverse of Symbolic Matrix

Compute Inverse of Hilbert Matrix with Symbolic Numbers

Compute Inverse of Block Matrix

Compute Inverse of Matrix Polynomial

Input Arguments

A—Input matrixsquare numeric matrix|square matrix of symbolic scalar variables|square symbolic matrix variable|square symbolic matrix function|symbolic expression with square size

Tips

Version History

R2022a: Compute inverse of symbolic matrix functions

R2021b: Compute inverse of symbolic matrix variables

See Also

`A`—Input matrix
square numeric matrix|square matrix of symbolic scalar variables|square symbolic matrix variable|square symbolic matrix function|symbolic expression with square size