Inmathematics, thespectrum of amatrix is theset of itseigenvalues.^[1][2][3] More generally, if${\displaystyle T:V\to V}$ is alinear operator on anyfinite-dimensionalvector space, its spectrum is the set of scalars${\displaystyle \lambda }$ such that${\displaystyle T-\lambda I}$ is notinvertible. Thedeterminant of the matrix equals the product of its eigenvalues. Similarly, thetrace of the matrix equals the sum of its eigenvalues.^[4][5][6]From this point of view, we can define thepseudo-determinant for asingular matrix to be the product of its nonzero eigenvalues (the density ofmultivariate normal distribution will need this quantity).

In many applications, such asPageRank, one is interested in the dominant eigenvalue, i.e. that which is largest inabsolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.

LetV be a finite-dimensionalvector space over somefieldK and supposeT :V →V is a linear map. Thespectrum ofT, denoted σ_T, is themultiset ofroots of thecharacteristic polynomial ofT. Thus the elements of the spectrum are precisely the eigenvalues ofT, and the multiplicity of an eigenvalueλ in the spectrum equals the dimension of thegeneralized eigenspace ofT forλ (also called thealgebraic multiplicity ofλ).

Now, fix abasisB ofV overK and supposeM ∈ Mat_K(V) is a matrix. Define the linear mapT :V →V pointwise byTx =Mx, where on the right-hand sidex is interpreted as a column vector andM acts onx bymatrix multiplication. We now say thatx ∈V is aneigenvector ofM ifx is an eigenvector ofT. Similarly, λ ∈K is an eigenvalue ofM if it is an eigenvalue ofT, and with the same multiplicity, and the spectrum ofM, written σ_M, is the multiset of all such eigenvalues.

Theeigendecomposition (or spectral decomposition) of adiagonalizable matrix is adecomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

Thespectral radius of asquare matrix is the largest absolute value of its eigenvalues. Inspectral theory, the spectral radius of abounded linear operator is thesupremum of the absolute values of the elements in the spectrum of that operator.

• Golub, Gene H.; Van Loan, Charles F. (1996),Matrix Computations (3rd ed.), Baltimore:Johns Hopkins University Press,ISBN 0-8018-5414-8
• Herstein, I. N. (1964),Topics In Algebra, Waltham:Blaisdell Publishing Company,ISBN 978-1114541016
• Kreyszig, Erwin (1972),Advanced Engineering Mathematics (3rd ed.), New York:Wiley,ISBN 0-471-50728-8
• Nering, Evar D. (1970),Linear Algebra and Matrix Theory (2nd ed.), New York:Wiley,LCCN 76091646

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