Manuals/calci/EIGENVALUES

EIGENVALUES (Matrix)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Matrix} is the array of numeric elements.

Description

• This function shows the Eigen value of the given matrix.
• In Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle EIGENVALUES (Matrix)} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Matrix} is any matrix values.
• In linear algebra,an eigen vector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it.
• Let A be a linear transformation represented by a matrix A.
• Let A is an nxn matrix,v is a non zero nx1 vector and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} is a scalar which may be either real or complex.
• Any value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} for which this equation has a solution is known as an eigenvalue of the matrix A.
• It is sometimes also called the characteristic value.
• The vector, v, which corresponds to this value is called an eigenvector.
• The eigenvalue problem can be rewritten as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A-\lambda.I).v=0} .
• If v is non-zero, this equation will only have a solution if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A-\lambda.I|.v=0} .
• This equation is called the characteristic equation of A, and is an nth order polynomial in $\lambda$ with n roots.
• These roots are called the eigenvalues of A.

Examples

1. EIGENVALUES([[3,7,5],[10,12,8],[6,8,14]])

Result

-2.018987498930866	25.303239119591886	5.715748379338994
-0.8195524172935329 0.3557792393359474 0.2128903683040517	0.5726193656991498 0.663334322125492 0.6212592923173481	0.02099755544415341 0.6583378387635402 -0.7541316747045657

2. EIGENVALUES([[5,6],[9,-11]])

Result

-13.862780491200214	7.8627804912002155
0.3031213645114406 0.9025310769284506	-0.9529519601620652 0.43062472662211493

See Also

• ANTIDIAGONAL
• CONFERENCE
• PASCAL

References

• Eigen Values

• List of Main Z Functions

• Z3 home