Einstein, Albert 157

Albert Einstein, an eminent mathematical physicist of the 20th

century, revolutionized our understanding of space, matter, and

time through his theories of special and general relativity. (Photo

courtesy of Topham/The Image Works)

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and set Bto be the matrix whose columns are the cor-

responding eigenvectors of A. Then it is possible to

show that:

A= BDB–1

This observation allows one to compute high powers of

Awith very little work. For instance, the quantity A100,

for instance, is just a product of three matrices:

A100 = (BDB–1)100

= BD100B–1

noting that D100 is simply:

This work also allows mathematicians to define the

square root of a matrix:

(provided the square roots of the eigenvalues are

defined), or to define new quantities, such as the loga-

rithm of a matrix:

Such actions have proved useful in the study of theoret-

ical physics and engineering.

The prefix eigen is German for “characteristic” or

“own.” The eigenvalues and eigenvectors of a matrix

completely characterize the matrix.

See also C

AYLEY

-H

AMILTON THEOREM

;

INVERSE

MATRIX

;

LINEARLY DEPENDENT AND INDEPENDENT

.

Einstein, Albert (1879–1955) German Relativity,

Quantum mechanics Born on March 14, 1879, in

Ulm, Germany, Albert Einstein is recognized as an

outstanding mathematical physicist whose work on

the special and general theories of relativity com-

pletely revolutionized how scientists think about

space, matter, and time. Although he regarded himself

as a physicist, Einstein’s work inspired many signifi-

cant developments in modern mathematics, including

the development of

TENSOR

analysis as the appropri-

ate means to describe the curvature of space.

The classical school environment did not suit Ein-

stein well. In 1895 he failed the entrance exam for a

Swiss technical school, where he hoped to study electri-

cal engineering. After attending a second school at

Aarau, he did eventually manage to enter the Zurich